lie algebras - lmuwehler/ss2018... · 2019. 1. 16. · chapter 1 matrix functions the paradigm of...

Joachim Wehler

Lie Algebras

DRAFT, Release 1.23

January 16, 2019

2

I have prepared these notes for the students of my lectures and my seminar. Thelectures and the seminar took place during the summer semesters 2017 and 2018(repetition) and the winter semester 2018/19 at the mathematical department ofLMU (Ludwig-Maximilians-Universitat) at Munich. I thank the participants of theseminar for making some simplifications and corrections.

Compared to the oral lecture in class these written notes contain some additionalmaterial.

Please report any errors or typos to [email protected]

Release notes:

• Release 1.23: Overview in Part II added.• Release 1.22: Chapter 8: Added Lemma 8.17.• Release 1.21: Chapter 8: Minor revisions.• Release 1.20: Chapter 8: Minor revisions.• Release 1.19: Chapter 8: Added proof of Theorem 8.8, minor revisions.• Release 1.18: Chapter 7: Proposition 7.6 added.• Release 1.17: Chapter 8: Minor revisions.• Release 1.16: Minor revisions.• Release 1.15: Update Chapter 6.• Release 1.14: Update Chapter 5, minor additions.• Release 1.13: Update Chapter 5, Section 5.2.• Release 1.12: Update Chapter 5, minor corrections and additions.• Release 1.11: Figure 8.1 corrected• Release 1.10: Added Chapter 8• Release 1.9: Added Chapter 7• Release 1.8: Added Chapter 6• Release 1.7: Added Chapter 5, minor corrections.• Release 1.6: Added Chapter 4.• Release 1.5: Update References.• Release 1.4: Added Chapter 3.• Release 1.3: Added Chapter 2.• Release 1.2: Update Chapter 1.• Release 1.1: General revision Chapter 1.

Contents

Part I General Lie Algebra Theory

1 Matrix functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1 Power series of matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Jordan decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.3 Exponential of matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2 Fundamentals of Lie algebra theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.1 Definitions and first examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.2 Lie algebras of the classical groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.3 Topology of the classical groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3 Nilpotent Lie algebras and solvable Lie algebras . . . . . . . . . . . . . . . . . . . 653.1 Engel’s theorem for nilpotent Lie algebras . . . . . . . . . . . . . . . . . . . . . . 653.2 Lie’s theorem for solvable Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . 753.3 Semidirect product of Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4 Killing form and semisimple Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . 914.1 Traces of endomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 914.2 Fundamentals of semisimple Lie algebras . . . . . . . . . . . . . . . . . . . . . . 964.3 Weyl’s theorem on complete reducibility of representations . . . . . . . 106

Part II Complex Semisimple Lie Algebras

5 Cartan decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1275.1 Toral subalgebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1275.2 sl(2,C)-modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1305.3 Cartan subalgebra and root space decomposition . . . . . . . . . . . . . . . . 140

6 Root systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1496.1 Abstract root system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1496.2 Action of the Weyl group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

v

vi Contents

6.3 Coxeter graph and Dynkin diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

7 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1837.1 The root system of a complex semisimple Lie algebra . . . . . . . . . . . . 1837.2 Root systems of the A,B,C,D-series . . . . . . . . . . . . . . . . . . . . . . . . . . . 1987.3 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

8 Irreducible Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2198.1 The universal enveloping algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2198.2 General weight theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2368.3 Finite-dimensional irreducible representations . . . . . . . . . . . . . . . . . . 246

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

Part IGeneral Lie Algebra Theory

Chapter 1Matrix functions

The paradigm of Lie algebras is the vector space of matrices with the commutator oftwo matrices as Lie bracket. These concrete examples even cover all abstract finitedimensional Lie algebra which are the focus of these notes. Nevertheless it is usefulto consider Lie algebras from an abstract viewpoint as a separate algebraic structurelike groups or rings.

If not stated otherwise, we denote by K either the field R of real numbers or thefield C of complex numbers. Both fields have characteristic 0. The relevant differ-ence is the fact that C is algebraically closed, i.e. each polynomial with coefficientsfrom K of degree n has exactly n complex roots. This result allows to transformmatrices over C to certain standard forms by transformations which make use of theeigenvalues.

1.1 Power series of matrices

At highschool every pupil learns the equation

exp(x) · exp(y) = exp(x+ y)

This formula holds for all real numbers x,y. Then exponentiation defines a map

exp : R→ R∗.

Students who attended a class on complex analysis will remember that expo-nentiation is defined also for complex numbers. Hence exponentiation extends to amap

exp : C→ C∗.

This map satisfies the same functional equation.The reason for the seamless transition from the real to the complex field is the

fact that the exponential map is defined by a power series

3

4 1 Matrix functions

exp(z) =∞

∑ν=0

1ν!· zν

which converges not only for all real numbers but for all complex numbers too.

In order to generalize further the exponential map we try to exponentiate argu-ments which are strictly upper triangular matrices.

Example 1.1 (Exponential of strictly upper triangular matrices). We denote by

n(3,K) :=

0 ∗ ∗

0 0 ∗0 0 0

∈M(3×3,K)

=(ai j)i j ∈M(n×n,K) : ai j = 0 if i ≥ j

the K-algebra of strictly upper triangular matrices. Matrices from n(3,K) can beadded and multiplied by each other, and they can be multiplied by scalars from thefield K. In particular, we consider the matrices

P =

0 p 00 0 00 0 0

,Q =

0 0 00 0 q0 0 0

∈ n(3,K).

We compute

exp(P)=∞

∑ν=0

1ν!·Pν =1+P=

1 p 00 1 00 0 1

,exp(Q)=∞

∑ν=0

1ν!·Qν =1+Q=

1 0 00 1 q0 0 1

.

Note Pν = Qν = 0 for all ν ≥ 2. Hence we do not need to care about questions ofconvergence.

i) To test the functional equation we compute on one hand

exp(P) · exp(Q) =

1 p 00 1 00 0 1

·1 0 0

0 1 q0 0 1

=

1 p pq0 1 q0 0 1

= 1+P+Q+PQ.

ii) On the other hand we compute

exp(P+Q).

For the computation of matrix products it is often helpful to introduce the basis (Ei j)1≤i, j≤nof the K-vector space M(n×n,K) with the matrices

Ei j ∈M(n×n,K)

having component = 1 at position with index i j and component = 0 at all otherpositions. Then

1.1 Power series of matrices 5

Ei j ·Ekm = δim.

We have

(P+Q)2 = (pE12 +qE23) · (pE12 +qE23) = pqE13 =

0 0 pq0 0 00 0 0

= PQ

and(P+Q)ν = 0, ν ≥ 3.

We obtain

exp(P+Q)=1+P+Q+1/2 ·(P+Q)2 =1+P+Q+(1/2)PQ=

1 p (pq)/20 1 q0 0 1

.

iii) Apparently, the functional equation is not satisfied:

exp(P+Q) 6= exp(P) · exp(Q).

Fortunately, the defect can be repaired by introducing as correction term the matrix

C := 1/2 · [P,Q] =

0 0 (pq)/20 0 00 0 0

=pq2

E13 = (1/2)PQ.

Here[P,Q] := PQ−QP

denotes the commutator of the matrices P and Q. Computing

PQ = pE12 ·qE23 = pqE13, QP = qE23 ·E13 = 0

we obtain[P,Q] = pqE13

(P+Q+C)2 =(pE12+qE23+(pq/2)E13)·(pE12+qE23+(pq/2)E13)= pqE13 = 2C

and(P+Q+C)ν = 0, ν ≥ 3.

Hence

exp(P+Q+1/2 · [P,Q]) = exp(P+Q+C) = 1+(P+Q+C)+(1/2)(P+Q+C)2

= 1+P+Q+2C = 1+P+Q+PQ =

0 p pq0 0 q0 0 0

= exp(P) · exp(Q).

iv) More general one can show: The exponential map can be defined on the strictupper triangular matrices


exp : n(3,K)→ GL(3,K).

It satisfies the functional equation in the generalized form: For A, B ∈M(n×n,K))

exp(A) · exp(B) = exp(A+B+(1/2)[A,B])

The goal of the present section is to extend the exponential map to all matrices.Therefore we need a concept of convergence for sequences and series of matrices.The basic ingredience is the norm of a matrix.

Definition 1.2 (Operator norm). We consider the K-vector space Kn with the Eu-clidean norm

‖x‖ :=

√n

∑i=1|xi|2 f or x = (x1, ...,xn) ∈Kn.

On the K-algebra M(n×n,K) we introduce the operator norm

‖A‖ := sup‖Ax‖ : x ∈Kn and ‖x‖ ≤ 1.

Note that ‖A‖< ∞ due to compactness of the unit ball

x ∈Kn : ‖x‖ ≤ 1.

Intuitively, the operator norm of A is the volume to which the linear map determinedby A blows-up or blows-down the unit ball of Kn.

The operator norm ‖A‖ does only depend on the endormorphism which is repre-sented by the matrix A. Hence, all matrices which represent a given endormorphismwith repect to different bases have the same operator norm.

The K-vector space M(n× n,K) of all matrices with components from K is anassociative K-algebra with respect to the matrix product

A ·B ∈M(n×n,K)

because(A ·B) ·C = A · (B ·C)

for matrices A,B,C ∈M(n×n,K).

Proposition 1.3 (Normed associative matrix algebra). The matrix algebra (M(n×n,K),‖‖)is a normed associative algebra:

1. ‖A‖= 0 iff A = 0


2. ‖A+B‖ ≤ ‖A‖+‖B‖

3. ‖λ ·A‖= |λ | · ‖A‖ with λ ∈ R

4. ‖A ·B‖ ≤ ‖A‖ · ‖B‖

5. ‖1‖= 1 with 1 ∈M(n×n,K) the unit matrix.

6. For a matrix A = (ai, j)i, j ∈M(n×n,K) holds

‖A‖sup ≤ ‖A‖ ≤ n · ‖A‖sup.

with‖A‖sup := sup|ai j| : 1≤ i, j ≤ n.

Proof. ad 6) For j = 1, ...,n denote by e j the j-th canonical basis vector of Kn. Thenfor all i = 1, ...,n

‖A‖ ≥ ‖Ae j‖=

√n

∑k=1|ak j|2 ≥ |ai j|,

hence‖A‖ ≥ ‖A‖sup.

Concerning the other estimate, the inequality of Cauchy Schwarz

|< x,y > |2 ≤ ‖x‖2 · ‖y‖2

implies for all x ∈Kn, in particular for |x| ≤ 1

‖Ax‖2 =n

∑i=1|

n

∑j=1

ai j · x j|2 ≤n

∑i=1

(n

∑j=1|ai j|2 ·

n

∑j=1|x j|2

)

≤ ‖x‖2 ·n

∑i=1

(n

∑j=1

supi, j|ai j|2

)= n2 · ‖x‖2‖A‖sup

2,

hence‖A‖ ≤ n · ‖A‖sup, q.e.d.

Remark 1.4 (Equivalence of norms).

1. Because the operator norm ‖A‖ and the sup-norm refering to the entries of A canbe mutually estimated, the following two convergence concepts are equivalentfor a sequence (Aν)ν∈N of matrices

Aν ∈M(n×n,K),ν ∈ N,


and a matrix A ∈M(n×n,K):

• limν→∞‖Aν −A‖= 0, i.e. lim

ν→∞Aν = A (norm convergence).

• The sequence (Aν)ν∈N converges to A component-by-component.

2. In particular each Cauchy sequence of matrices is convergent: The matrix algebra

(M(n×n,K),‖‖)

is complete, i.e. a Banach algebra.

3. The vector space M(n× n,K) is finite dimensional, its dimension is n2. Henceany two norms are equivalent in the sense that they dominate each other. There-fore the structure of a topological vector space on M(n×n,K) does not dependon the choice of the norm.

Because (M(n×n,K),‖‖) is a normed algebra according to Proposition 1.3, con-cepts from analysis like convergence, Cauchy sequence, continuous function, andpower series also apply to matrices. In particular, the commutator is a continuousmap: For A, B ∈M(n×n,K)

‖[A,B]‖= ‖AB−BA‖ ≤ ‖AB‖+‖BA‖ ≤ 2‖A‖‖B‖.

We recall the fundamental properties of a complex power series

∞

∑ν=0

cν · zν

with radius of convergence R > 0. Set

∆(R) := z ∈ C : |z|< R.

Then

• The series is absolutely convergent in ∆(R), i.e. ∑∞ν=0 |cν | · |z|ν is convergent

for z ∈ ∆(R).

• The series converges compact in ∆(R), i.e. the convergence is uniform on eachcompact subset of ∆(R).

• The series is infinitely often differentiable in ∆(R), its derivation is obtained bytermwise differentiation.


Lemma 1.5 (Power series of matrices). Consider a power series

f (z) =∞

∑ν=0

cν · zν

with coefficients cν ∈K,ν ∈ N, and radius of convergence R > 0. Let

B(R) := A ∈M(n×n,K) : ‖A‖< R

be the open ball in M(n×n,K) around zero with radius R.

For a matrix A ∈ B(R) then:

• The series

f (A) :=∞

∑ν=0

cν ·Aν := limn→∞

(n

∑ν=0

cν ·Aν

)∈M(n×n),K)

is convergent, absolutely convergent and compact convergent in B(R).

• The series f (A) satisfies [ f (A),A] = 0 with the commutator

[ f (A),A] := f (A) ·A−A · f (A).

• The functionf : B(R)→M(n×n,K), A 7→ f (A),

is continuous.

Proof. i) We apply the Cauchy criterion: For N > M holds

‖N

∑ν=0

cν ·Aν −M

∑ν=0

cν ·Aν‖ ≤N

∑ν=M+1

|cν | · ‖A‖ν .

If r := ‖A‖< R then ∑∞ν=0 |cν | ·rν converges. Hence the Cauchy criterion is satisfied

and the limit

limn→∞

(n

∑ν=0

cν ·Aν

)exists. The remaining statements about convergence follow from the estimate

‖ f (A)‖ ≤∞

∑ν=0|cν | · ‖A‖ν

and the corresponding properties of complex power series.


ii) The equation [A, f (A)] = 0 about the commutator follows: We take the limit limn→∞

and employ the continuity of the commutator[A,

∞

∑ν=0

cν ·Aν

]= lim

n→∞

[A,

n

∑ν=0

cν ·Aν

]= lim

n→∞

n

∑ν=0

cν [A, ·Aν ] = 0

iii) Continuity of f is a consequence of the compact convergence, q.e.d.

Definition 1.6 (Derivation of matrix functions). Let I ⊂ R be an open interval. Amatrix function

A : I→M(n×n,K)

is differentiable at a point t0 ∈ I iff the limit

limh→0

A(t0 +h)−A(t0)h

=: A′(t0) ∈M(n×n,K)

exists. In this case we employ the notation

dAdt

(t0) := A′(t0).

Lemma 1.7 (Derivation of power series of matrices depending on a parameter).Let I ⊂ R be an open interval.

i) Consider a complex power series

f (z) =∞

∑ν=0

cν · zν

with radius of convergence R > 0. and

A : I→ B(R)⊂M(n×n,K)

a differentiable function, which satisfies for all t ∈ I

[A′(t),A(t)] = 0.

Then also the function

f A : I→M(n×n,K), t 7→ f (A(t)) :=∞

∑ν=0

cν ·Aν(t),

is differentiable for all t ∈ I and satisfies the chain rule


ddt

f (A(t)) = f ′(A(t)) ·A′(t) = A′(t) · f ′(A(t)).

Here f ′ denotes the derivation of the complex power series f term by term.

ii) Consider two differentiable maps

A, B : I −→M(n×n,K).

Then for all t ∈ I holds the product rule

ddt

A(t) ·B(t) = A′(t) ·B(t)+A(t) ·B′(t).

Proof. i) The series

f (A(t)) =∞

∑ν=0

cν ·Aν(t)

and

f ′(A(t)) =∞

∑ν=1

ν · cν ·Aν−1(t)

are convergent because ‖A(t)‖< R. We have

f ′(A(t)) ·A′ =

(∞

∑ν=1

ν · cν ·Aν−1(t)

)·A′.

Due to the compact convergence we may differentiate f (A(t)) term by term. Theproof of the lemma reduces to proving

ddt

Aν(t) = ν ·Aν−1(t) ·A′(t) = ν ·A′(t) ·Aν−1(t).

Here the proof goes by induction on ν ∈N. For the induction step assume the claimto hold for ν ∈ N. Analogous to the proof of the product rule from calculus by cleverinsert of a suitable term:

ddt

Aν+1(t) = limh→0

Aν+1(t +h)−Aν+1(t)h

= limh→0

Aν(t +h) ·A(t +h)−Aν(t) ·A(t)h

=

= limh→0

Aν(t +h) ·A(t +h)−Aν(t) ·A(t +h)+Aν(t) ·A(t +h)−Aν(t) ·A(t)h

=

= limh→0

Aν(t +h) ·A(t +h)−Aν(t) ·A(t +h)h

+ limh→0

Aν(t) ·A(t +h)−Aν(t) ·A(t)h

=

= limh→0

(Aν(t +h)−Aν(t)

h·A(t +h)

)+ lim

h→0

(Aν(t) ·

A(t +h)−A(t)h

)=


=

(ddt

Aν(t)

)·A(t)+Aν(t) ·A′(t) = ν ·Aν−1(t) ·A′(t) ·A(t)+Aν(t) ·A′(t) =

(ν +1) ·Aν(t) ·A′(t) = (ν +1) ·A′(t) ·Aν(t).

To obtain the third last equation we have employed the induction assumption. Thesecond last equation and the last equation follow from the precondition [A′(t),A(t)] = 0.

ii) The proof is similar to part i):

limh→0

A(t +h) ·B(t +h)−A(t) ·B(t)h

=

= limh→0

A(t +h) ·B(t +h)−A(t) ·B(t +h)+A(t) ·B(t +h)−A(t)B(t)h

=

limh→0

A(t +h)−A(t)h

· limh→0

B(t +h)−A(t) · limh→0

B(t +h)−B(t)h

q.e.d.

1.2 Jordan decomposition

The central aim of Jordan theory is to decompose a complex vector space V into sub-spaces which are eigenspaces or generalized eigenspaces of a given endomorphismf ∈ End(V ) and to decompose f as the sum of two more specific endomorphisms.

Definition 1.8 (Eigenspaces and generalized eigenspaces). Consider a K-vectorspace V , a fixed endomorphism f ∈ End(V ), and λ ∈K.

• IfVλ ( f ) := ker( f −λ ) 6= 0

then Vλ ( f ) is the eigenspace of f with respect to λ .

• IfV λ ( f ) :=

⋃n∈N

ker( f −λ )n 6= 0

then V λ ( f ) is the generalized eigenspace of f with respect to λ .

Remark 1.9 (Stableness of generalized eigenspaces).

1.2 Jordan decomposition 13

1. All generalized eigenspaces V λ ( f ) are f -invariant, i.e.

f (V λ ( f ))⊂V λ ( f ),

because[( f −λ )n, f ] = 0.

2. Every non-zero generalized eigenspace V λ ( f ) contains at least one eigenvector vof f with eigenvalue λ : Take a non-zero vector v0 ∈ V λ ( f ) and choose n ∈ Nmaximal with

v := ( f −λ )n(v0) 6= 0.

We recall the following matrix representation of endomorphisms.

Definition 1.10 (Important types of endomorphisms). Consider an n-dimensionalK-vector space V .

1. An endomorphism f ∈ End(V ) is triangularizable iff V has an f -stable flag, i.e.a sequence (Vi)i=0,...,n of subspaces

Vi (Vi+1, i = 0, ...,n−1

exists satisfying f (Vi)⊂Vi, i = 0, ...,n.

2. An endomorphism f ∈ End(V ) is diagonalizable iff it can be represented by adiagonal matrix from M(n× n,K) . If the base fields is C then semisimple is asynonym for diagonalisable.

3. An endomorphism f ∈ End(V ) is nilpotent iff f k(V ) = 0 for a suitable k ∈ N.

A matrix A = (ai j) ∈ M(n× n,K) is an upper triangular matrix iff ai j = 0 forall i > j. Apparently an endomorphism is triangularizable iff it can be representedby an upper triangular matrix from M(n×n,K).

Note. The only endomorphism f ∈ End(V ), which is both diagonalizable andnilpotent, is f = 0.

Whether an endomorphism f is triangularizable or even diagonalizable dependson the roots of its characteristic polynomial. These roots are the eigenvalues of f .

Definition 1.11 (Characteristic polynomial of an endomorphism). Denote by Va K-vector space and by f ∈ End(V ) an endomorphism. The characteristic polyno-mial of f is the polynomial

pchar(T ) := det(T − f ) ∈K[T ].

The latter is defined as


det(T ·1−A) ∈K[T ]

with 1 ∈M(n×n,K) the unit matrix and A ∈M(n×n,K) a matrix representing f .The resulting polynomial is independent from the representing matrix. For a root λ ∈Kwe denote by

µ(pchar;λ ) ∈ N

the multiplicity of the root, i.e. the algebraic multiplicity of λ .

The criteria for respectively triangularization and diagonalization of an endomor-phism are stated in the following Proposition 1.13 and Proposition 1.12.

Proposition 1.12 (Triangular form). For an endomorphism f ∈ End(V ) with afinite dimensional K-vector space V are equivalent:

1. The endomorphism f is triangularizable.

2. The characteristic polynomial pchar splits over K into linear factors.

For the proof see [12].

In particular, over K= C every endomorphism is triangularizable.

Proposition 1.13 (Diagonal form). For an endomorphism f ∈ End(V ) with a finitedimensional K-vector space V are equivalent:

1. The endomorphism f is diagonalizable.

2. The characteristic polynomial pchar splits over K into linear factors, and for alleigenvalues λ of f the algebraic multiplicity equals the geometric multiplicity,i.e.

µ(pchar;λ ) = dim Vλ ( f )(Geometric multiplicity).

3. The vector space V splits as direct sum of eigenspaces

V =⊕

λ eigenvalue

Vλ ( f ).

For the proof see [12].

Lemma 1.14 (Diagonal approximation). Consider a complex, finite dimensionalvector space V . For each endomorphism f ∈ End(V ) a sequence ( fν)ν∈N of diago-nalizable endomorphisms fν ∈ End(V ) exists with

f = limν→∞

fν .


Proof. Consider an endomorphism f ∈ End(V ). Because the base field C is alge-braically closed the characteristic polynomial pchar(T ) ∈ C of f splits into linearfactors. According to Proposition 1.12 the endomorphism f can be represented byan upper triangular matrix A

A = ∆ +N

with a diagonal matrix ∆ = diag(λ1, ...,λn) and a strictly upper triangular matrix

N = (ai j),ai j = 0 if i≥ j.

For i = 1, ...,n define successively sequences ai = (aiν)ν∈N of complex numbers

converging to zero such that the sets

λ j +a jν : j = 1, ..., i−1 and ν ∈ N

andλi +ai

ν : ν ∈ N

are disjoint. Such choice is possible because the field C is uncountably infinite. Foreach ν ∈ N define

Aν := diag(λ1 +a1ν , ...,λn +an

ν)+N.

Each matrix Aν has eigenvalues

λi +aiν , i = 1, ...,n

and for each fixed ν ∈ N all eigenvalues of Aν are pairwise distinct. Hence Aν

represents a diagonalizable endomorphism fν due to Proposition 1.13. Moreover

A = limν→∞

Aν ,

i.e.f = lim

ν→∞fν ,

q.e.d.

Theorem 1.15 (Jordan decomposition). Let f ∈ End(V ) be an endomorphism ofa finite dimensional complex vector space V .

1. A unique decomposition

f = fs + fn (Jordan decomposition)

exists with a semisimple endomorphism fs ∈ End(V ) and a nilpotent endomor-phism fn ∈ End(V ) such that both satisfy

[ fs, fn] = 0.


2. The components fs and fn depend on f in a polynomial way, i.e. polynomials

ps(T ), pn(T ) ∈ C[T ]

exist with ps(0) = pn(0) = 0 such that

fs = ps( f ) and fn = pn( f ).

In particular, if [ f ,g] = 0 for an endomorphism g ∈ End(V ) then

[ fs,g] = [ fn,g] = 0.

3. The vector space V splits as direct sum of the generalized eigenspaces

V =⊕

λ eigenvalue

V λ ( f ).

For each eigenvalue λ the generalized eigenspace of f equals the eigenspaceof fs:

V λ ( f ) =Vλ ( fs).

Proof. W.l.o.g. V 6= 0. Also w.l.o.g. f 6= 0 because the decomposition

0 = fs + fn

implies fs = fn. Zero is the only endomorphism which is both semisimple and nilpo-tent. We obtain

fs = fn = 0.

i) Minimal polynomial: Set R := C[T ]. The family ( f k)k∈N is linearly dependentbecause End(V ) is a complex vector space with finite dimension (dim V )2. There-fore an equation exists

f k =k−1

∑i=0

αi · f i, αi ∈ C.

As a consequence a non-zero polynomial p ∈ R exists with

p( f ) = 0 ∈ End(V ).

The ring R is a principal ideal domain. Hence the non-zero ideal

< p ∈ R : p( f ) = 0 ∈ End(V )>

formed by all polynomials, which annihilate the endomorphism f , is generated by aunique monic polynomial of positive degree with leading coefficient = 1, the mini-mal polynomial of f

pmin(T ) ∈ R.


ii) Splitting idV as a sum of pairwise commuting projectors: Because C is alge-braically closed, pmin(T ) splits into linear factors with positive exponents mi ∈ N∗

pmin(T ) =r

∏i=1

(T −λi)mi

with λi 6= λ j if i 6= j.

For i = 1, ...,r consider the polynomials

pi :=pmin(T )

(T −λi)mi∈ R.

Because R is a principal ideal domain, a polynomial g ∈ R exists with

< pi : i = 1, ...,r >=< g > .

The only common factor of all pi are the units of R, hence w.l.o.g we have g = 1.Therefore, for i = 1, ...,r polynomials ri ∈ R exist with

1 =r

∑i=1

ri · pi ∈ R.

For i = 1, ...,r we define the endomorphisms

Ei := ri( f ) · pi( f ) ∈ End(V ).

They satisfy:

•idV =

n

∑i=1

Ei

• If i 6= j then pmin dividesri · pi · r j · p j ∈ R.

Therefore pmin( f ) = 0 ∈ End(V ) implies

Ei ·E j = 0 ∈ End(V ).

• The family (Ei)i=1,...,r is a family of projectors:

E2k = Ek

r

∑i=1

Ei = Ek idV = Ek.

Hence the family (Ei)i=1,...,r splits the identity idV as a sum of pairwise commutingprojectors. We define the corresponding subspaces

Vi := im Ei ⊂V, i = 1, ...,r


and obtain

V =r⊕

i=1

Vi.

For all i = 1, ...,r by definition of Ei as a polynomial in f

[ f ,Ei] = 0,

which implies that the subspace Vi is f -stable.

iii) Definition of fs and fn: On each subspace

Vi, i = 1, ...,r,

the corresponding linear map Ei - being a projector - acts as identity, hence theendomorphism

λi · (Ei|Vi) ∈ End(Vi)

acts as multiplication by λi. We define the polynomial

ps(T ) :=r

∑i=1

λi · ri(T ) · pi(T ) ∈ R

and the corresponding endomorphism

fs := ps( f ) =r

∑i=1

λi ·Ei ∈ End(V ).

Then fs ∈ End(V ) is semisimple with eigenspaces

Vi =Vλi( fs).

For the nilpotent component of f we consider the polynomial

pn(T ) := T − ps(T ) ∈ C[T ]

and define the endomorphism

fn := pn( f ) = f − fs ∈ End(V ).

Thenf = fs + fn.

The two definitionsfs := ps( f ), fn := pn( f )

imply[ f , fs] = [ f , fn] = 0.


In order to prove the nilpotency of fn we consider the restrictions to an arbitrarybut fixed subspace Vi and set m := mi. On Vi

fn = f − fs = ( f −λi)− ( fs−λi).

Using [ fs, f ] = 0 the binomial theorem implies

f mn =

m

∑µ=0

(mµ

)(−1)m−µ( f −λi)

µ ( fs−λi)m−µ .

Here the second factor( fs−λi)

m−µ

vanishes for 0≤ µ < m. The first factor

( f −λi)µ

vanishes for µ = m, because for v ∈Vi

( f−λi)m(v)= ( f−λi)

m(Ei(v))= ( f−λi)m(ri( f ) pi( f ))(v)= (ri( f ) pmin( f ))(v)= 0,

i.e.Vi ⊂V λi( f ).

As a consequence of the binomial theorem

f min (Vi) = 0.

Hence the restriction fn|Vi is nilpotent for every i = 1, ...,r, which implies the nilpo-tency of fn ∈ End(V ).

iv) V λi( f )⊂Vi: Consider an arbitrary, but fixed i = 1, ...,r. We use the direct sumdecomposition from part ii)

V =r⊕

j=1

Vj.

Consider v ∈V λi( f )⊂V and decompose

v =r

∑j=1

v j,v j ∈Vj for j = 1, ...,r.

For large n ∈ N

0 = ( f −λi)n(v) =

r

∑j=1

( f −λi)n(v j).

The f -stableness of each Vj implies

( f −λi)n(v j) ∈Vj.


Consider an index j = 1, ...,r such that v j 6= 0. Then

( f −λi)n(v j) = 0,

i.e. f has the generalized eigenvector v j for λi:

v j ∈V λi( f ).

Then f has also an eigenvector w ∈Vλi( f ) with eigenvalue λi: For suitable k ∈ N

w := ( f −λi)k(v j) 6= 0 but ( f −λi)(w) = 0.

The f -stableness of Vj implies w ∈Vj. We obtain for large m

0 = ( f −λ j)m(w) = (λi−λ j)

m ·w.

Here the left equality is due to

w ∈Vj ⊂V λ j( f )

with the last inclusion proven in part iii). And the right equality is due to

w ∈Vλi( f ).

The equality0 = (λi−λ j)

m ·w

implies j = i. As a consequence v = vi ∈Vi, which finishes the proof of V λi( f )⊂Vi.

Together with the opposite inclusion from part iii) we obtain

Vi =V λi( f )

and

V ∼=r⊕

i=1

Vi ∼=r⊕

i=1

V λi( f )∼=r⊕

i=1

Vλi( fs).

v) The family (λi)i=1,...,r comprises exactly the eigenvalues of f : For each root λ

of the minimal polynomial pmin(T ) of f the generalized eigenspace V λ ( f ) containsan eigenvector of f with eigenvalue λ , see part iv) of the proof. Therefore any rootof pmin(T ) is an eigenvalue of f .

For the oppposite inclusion we consider an eigenvalue λ of f with correspondingeigenvector v:

f (v) = λ ·v.

Because pmin( f ) = 0 ∈ End(V ) we have in particular

pmin( f )(v) = 0 ∈V.


Hence0 = pmin( f )(v) = pmin(λ ) ·v ∈V.

Because v 6= 0 we concludepmin(λ ) = 0 ∈ C,

i.e. the eigenvalue λ is a root of the minimal polynomial.

vi) Uniqueness of the Jordan decomposition: Assume a second decomposition

f = f ′s + f ′n

with the properties stated in part 2) of the theorem. Then

[ f , f ′n] = [ f ′s , f ′n]+ [ f ′n, f ′n] = 0.

Fromfn = pn( f )

we obtain[ fn, f ′n] = 0.

Analogously, one proves[ fs, f ′s ] = 0.

As a consequencefs− f ′s = f ′n− fn ∈ End(V )

is an endomorphism which is both semisimple - being the sum of two commutingsemsisimple endomorphisms - and nilpotent - being the sum of two commutingnilpotent endomorphisms. Hence it is zero.

vii) Killing the constant terms: The polynomials ps and pn may be choosen with

ps(T ) = pn(0) = 0 :

On one hand, if λi = 0 for one index i = 1, ...,r then

V 0( f ) 6= 0

and the generalized eigenspace of f contains also an eigenvector of f witheigenvalue 0, i.e.

0 6= ker f .

Ifps(T ) = a0 +a1T + ...+akT k ∈ C[T ]

thenfs = ps( f ) = a0 · id +a1 · f + ...+ak · f k End(V ).

Restriction to


ker f ∩V 0( f ) 6= 0

shows a0 = 0. Hence ps(T ) ∈ C[T ] has no constant term, i.e. ps(0) = 0. From thedefinition pn(T ) := T − ps(T ) follows

pn(0) =−ps(0) = 0.

On the other hand, if λi 6= 0 for all indices i = 1, ...,r then pmin(0) 6= 0: Otherwise

pmin(T ) = T ·q(T ) with q ∈ C[T ],q 6= 0, deg q < deg pmin.

Because the minimal polynomial has minimal degree with respect to all non-zeropolynomials annihilating f , we get in particular

q( f ) 6= 0,

i.e. a vector v ∈V exists with q( f )(v) 6= 0. As a consequence

0 = pmin( f ) = f ·q( f )

implies0 6= q( f )(v) ∈ ker f ,

a contradiction, because λ = 0 is not an eigenvalue.

Now one replaces the polynomials ps and pn by

ps := ps−ps(0)

pmin(0)· pmin

and

pn := pn−pn(0)

pmin(0)· pmin.

Thenps(0) = pn(0)

without changingfs = ps( f ) and fn = pn( f ), q.e.d.

The Caleigh-Hamilton theorem can be obtained as a corollary of the Jordan de-composition.

Theorem 1.16 (Cayleigh-Hamilton). Consider a complex, finite-dimensional vec-tor space V and an endomorphism f ∈ End(V ).

Then the minimal polynomial pmin(T ) divides pchar(T ) in C[T ]. As a conse-quence, the characteristic polynomial pchar(T ) ∈ C annihilates f , i.e.

pchar( f ) = 0 ∈ End(V ).


Note that both polynomials pmin(T ) and pchar(T ) have the same roots.

Proof. Denote byf = fs + fn

the Jordan decomposition of f according to Theorem 1.15. We choose an arbitrarybut fixed j ∈ 1, ...,r and set

λ := λ j, m := m j, W :=V λ ( f ), and k := dim W.

Because W is f -stable, the Jordan decomposition of f restricts to the Jordan decom-position on W :

f ′ = f ′s + f ′n

Here priming indicates restriction to W . We have

f ′s = λ · idW .

The nilpotent endomorphism f ′n can be represented by a strictly upper triangularmatrix. Hence f ′ is represented by an upper triangular matrix

A ∈M(k× k,C)

with all diagonal elements equal to λ . The characteristic polynomial of f ′ is

pchar, f ′(T ) = det(T −A) = (T −λ )k ∈ C[T ].

The minimal polynomial of f ′ is

(T −λ )m,

i.e. each polynomial in f ′, which vanishes on W has degree at least m. Apparently

( f ′−λ )k = ( f ′n)k= 0.

Hencem≤ k.

Summing up, the characteristic polynomial of f is

pchar(T ) =r

∏j=1

(T −λ j)dim V λ j ( f )

and for each j = 1, ...,r holds

m j ≤ dim V λ j( f ).

In particular,pmin(T ) divides pchar(T ),


and both polynomials have the same roots. Because pmin( f ) = 0 also pchar( f ) = 0,q.e.d.

Apparently the statement pchar( f ) = 0 of the Cayleigh-Hamilton theorem alsoholds for an endomorphism of a real vector space V . Because any real endomor-phism f defines the complex endomomorphism

f ⊗ id

of the complexification V ⊗RC.

1.3 Exponential of matrices

The complex exponential series

exp(z) =∞

∑ν=0

zν

ν!

has convergence radius R = ∞. We now generalize the exponential series from com-plex numbers z ∈ C as argument to matrices A ∈M(n×n,K) as argument.

Definition 1.17 (Exponential of matrices). For any matrix A ∈ M(n× n,K) onedefines the exponential

exp(A) :=∞

∑ν=0

Aν

ν!∈M(n×n,K).

Proposition 1.18 (Derivation of the exponential). Consider an open interval I⊂Rand a differentiable function

A : I→M(n×n,K)

with [A′(t),A(t)] = 0 for all t ∈ I. Then for all t ∈ I

ddt(exp A(t)) = A′(t) · exp A(t) = exp A(t) ·A′(t).

Proof. We apply Lemma 1.7 with the power series

f (z) =∞

∑ν=0

1ν!· zν

1.3 Exponential of matrices 25

and its derivativef ′(z) = f (z).

We obtain

ddt(exp A(t)) =

(∞

∑ν=0

1ν!·Aν(t)

)·A′(t) = exp A(t) ·A′(t) = A′(t) · exp A(t).

Proposition 1.19 (Exponential of matrices). The exponential of matrices A,B ∈M(n×n,C)satisfies the following rules:

1. Functional equation in the commutative case: If [A,B] = 0 then

exp(A+B) = exp(A) · exp(B).

2. Transposition: (exp A)> = exp(A>)

3. Base change: If S ∈ GL(n,C) then

exp(S ·A ·S−1) = S · exp(A) ·S−1.

4. Determinant and trace: det(exp A) = exp(tr A).

Proof. ad 1: First we apply the binomial theorem with [A,B] = 0:

exp(A+B) =∞

∑ν=0

(A+B)ν

ν!=

∞

∑ν=0

1ν!

(ν

∑µ=0

(ν

µ

)Aν−µ ·Bµ

)=

=∞

∑ν=0

(ν

∑µ=0

Aν−µ

(ν−µ)!·

Bµ

µ!

)Secondly we invoke the Cauchy product formula. It rests on the fact that any rear-rangement of absolutely convergent series is admissible:

∞

∑ν=0

(ν

∑µ=0

Aν−µ

(ν−µ)!·

Bµ

µ!

)=

∞

∑ν=0

Aν

ν!·

∞

∑ν=0

Bν

ν!= exp(A) · exp(B).

ad 2: Taking limN→∞

of (N

∑ν=0

Aν

ν!

)>=

N

∑ν=0

(A>)ν

ν!

ad 3: Taking limN→∞

of

S ·

(N

∑ν=0

Aν

ν!

)·S−1 =

N

∑ν=0

S ·Aν ·S−1

ν!=

N

∑ν=0

(S ·A ·S−1)ν

ν!


ad 4: According to Proposition 1.12 and part 3) of the proposition w.o.l.g.

A =

λ1 ∗. . .

0 λn

, λi ∈ C, i = 1, ...,n,

is an upper triangular matrix. We obtain

Aν =

λ ν1 ∗

. . .0 λ ν

n

,ν ∈ N,

and conclude

det(exp A) = det

(∞

∑ν=0

Aν

ν!

)= det

eλ1 ∗. . .

0 eλn

.

Hence

det(exp A) =n

∏i=1

eλi = eλ1+...+λn = exp(tr A).

Corollary 1.20 (Exponential map). The exponential defines a map

exp : M(n×n,K)→ GL(n,K), A 7→ exp(A),

i.e. the matrix exp(A) is invertible, and exp(A)−1 = exp(−A).

The inverse of exponentiation is taking the logarithm. But even in the complexcase the exponential map is not injective because exp(z+ 2πi) = exp(z). Anyhowthe exponential map is locally invertible. And this property continues to hold for theexponential of matrices.

Recall the complex power series of the logarithm

log(1+ z) =∞

∑ν=1

(−1)ν+1

ν· zν ,z ∈ C,

and the complex geometric series

∞

∑ν=0

zν ,z ∈ C.

Both power series have radius of convergence R = 1.


Definition 1.21 (Logarithm and geometric series of matrices). For a matrix A ∈M(n×n,K)with ‖A‖< 1 one defines its logarithm

log(1+A) :=∞

∑ν=1

(−1)ν+1 · Aν

ν∈M(n×n,K)

and its geometric series∞

∑ν=0

Aν ∈M(n×n,K).

Proposition 1.22 (Derivation of logarithm).

1. For a matrix A ∈M(n×n,K) with ‖A‖< 1 the matrix 1−A is invertible with

(1−A)−1 =∞

∑ν=0

Aν .

2. Consider an open interval I ⊂ R and a differentiable function

B : I→M(n×n,K)

with ‖B(t)−1‖< 1 and [B′(t),B(t)] = 0 for all t ∈ I.

Then for all t ∈ I the inverse B(t)−1 exists and

ddt(log B(t)) = B(t)−1 ·B′(t) = B′(t) ·B(t)−1.

Proof. ad 1) For each n ∈ N

(1−A) ·n

∑ν=0

Aν =n

∑ν=0

Aν −n+1

∑ν=1

Aν = 1−An+1.

Because ‖A‖< 1 it follows

(1−A) ·∞

∑ν=0

Aν = 1− limn→∞‖A‖n+1 = 1.

ad 2) For t ∈ I defineA := 1−B(t).

Because ‖A‖= ‖1−B(t)‖< 1 apply part 1) to A:

B(t) = 1−A

has the inverse


B(t)−1 =∞

∑ν=0

Aν =∞

∑ν=0

(1−B(t))ν .

In addition,log(B(t)) = log(1+(B(t)−1))

is well-defined. Lemma 1.7 with the power series

f (1+ z) =∞

∑ν=1

(−1)ν+1

ν· zν

and its derivative

f ′(1+ z) =∞

∑ν=0

(−z)ν

implies

ddt(log B(t)) =

ddt

log(1+(B(t)−1)) =

(∞

∑ν=0

(1−B(t))ν

)·B′(t) =

= B(t)−1 ·B′(t) = B′(t) ·B(t)−1.

Proposition 1.23 (Exponential and logarithm as locally inverse maps). We have

1. log(exp A) = A for A ∈M(n×n,C) if ‖A‖< log 2

2. exp(log B) = B for B ∈M(n×n,C) if ‖B−1‖< 1 or (B−1) nilpotent.

Proof. ad 1) The assumption ‖A‖< log 2 implies

‖exp(A)−1‖= ‖∞

∑ν=1

Aν

ν!‖ ≤

∞

∑ν=1

‖A‖ν

ν!= exp(‖A‖)−1 < 2−1 = 1.

Hencelog(exp A) = log(1+(exp(A)−1))

is a well-defined convergent power series.In order to prove the equality

log(exp A) = A

we first consider a diagonal matrix

A =

λ1 0. . .

0 λn

.


Then

log(exp(A)) =

log(eλ1) 0. . .

0 log(eλn)

=

λ1 0. . .

0 λn

= A.

Here we use that the series of the logarithm log(eλi), i = 1, ...,n, computes the mainvalue of the logarithm.

According to Proposition 1.12 an arbitrary complex matrix A ∈ M(n× n,C) istriangularizable, i.e. an invertible matrix S ∈ GL(n,C) exists with

S ·A ·S−1 = B

an upper triangular matrix. According to Lemma 1.14 a series of diagonalizablematrices (Bν)ν∈N exists with

B = limν→∞

Bν .

For each ν ∈ N an invertible matrix Sν ∈ GL(n,C) exists with

Sν ·Bν ·S−1ν = ∆ν

a diagonal matrix. Because ‖A‖= ‖B‖ we may assume: For ν ∈N sufficiently largealso

‖∆ν‖= ‖Bν‖< log 2

andBν = S−1

ν ·∆ν ·Sν = S−1ν · log(exp ∆ν) ·Sν =

= log(exp(S−1ν ·∆ν ·Sν)) = log(exp Bν).

Taking the limit and using the continuity of the functions log and exp gives

B = log(exp(B))

and

A = S−1 ·B ·S = S−1 · log(exp(B)) ·S = log(exp(S−1 ·B ·S)) = log(exp(A)).

ad 2) By assumption

exp(log(B)) = exp(log(1+(B−1)))

is a convergent power series. The proof of the equality

exp(log B) = B

goes along the same line as the proof of part 1): First, one proves the equality fordiagonal matrices. Secondly, one approximates the triagonizable matrix B by a se-


quence of diagonalizable matrices. And finally one uses the continuity of functionswhich are given by power series, q.e.d.

Remark 1.24 (Counter example against invertibility). In Proposition 1.23, part 1)the assumption ‖A‖< log 2 cannot be dropped: Consider the matrix

A = 2π i ·1 ∈M(n×n,C)

with ‖A‖= 2π > log 2. We have

exp(A) = e2π i ·1= 1

hencelog(exp(A)) = log(1) = 0 6= A.

Theorem 1.25 (Surjectivity of the complex exponential map). The exponentialmap of complex matrices

exp : M(n×n,C)→ GL(n,C), A 7→ exp A,

is surjective.

Proof. Consider a fixed but arbitrary matrix B∈GL(n,C). According to Theorem 1.15the vector space Cn splits into the sum of the generalized eigenspaces with respectto the eigenvalues λ of B

Cn ∼=⊕

λ

V λ (B)

and each restriction Bλ := B|V λ (B) is an endomorphism of V λ (B).

W.l.o.g. we may assume B = Bλ with a fixed λ ∈ C∗. The matrix

N := B−λ ·1

is nilpotent and

B = λ ·1+N = λ (1+1λ·N).

Moreover

A1 := log(1+1λ·N) =

∞

∑ν=1

(−1)ν+1

ν· N

ν

λ ν

is well-defined because the sum is finite. We have

exp A1 = exp(log(1+1λ·N)) = 1+

1λ·N


according to Proposition 1.23. After choosing a complex logarithm µ ∈C with eµ = λ

and settingA := µ ·1+A1,

Proposition 1.19 implies

exp A = exp(µ ·1) · exp A1 = (λ ·1) · exp A1 = λ (1+1λ·N) = B, q.e.d.

Remark 1.26 (Counter examples against surjectivity). In general the exponentialmap is not surjective.

1. Complex case: Set

sl(2,C) := A ∈M(2×2,C) : tr A = 0

andSL(2,C) := B ∈ GL(2,C) : det B = 1

and considerexp : sl(2,C)→ SL(2,C).

The map is well-defined due to Proposition 1.19. The matrix

B =

(−1 b0 −1

)∈ SL(2,C),b ∈ C∗,

has no inverse image.

2. Real case: Set

GL+(2,R) := B ∈ GL(2,R) : det B > 0

and considerexp : M(2×2,R)→ GL+(2,R).

The matrix

B =

(−1 b0 −1

)∈ GL+(2,R), b ∈ R∗,

has no inverse image.

Proof. If a matrix A ∈ M(n× n,K) has the eigenvector v with eigenvalue λ thenthe matrix exp A has the same vector as eigenvector with eigenvalue eλ : Theequation A ·v = λ ·v and its iterates

Aν ·v = λν ·v,ν ∈ N,

imply

32 1 Matrix functions(n

∑ν=0

Aν

ν!

)·v =

(n

∑ν=0

λ ν

ν!

)·v ∈Kn.

Taking the limit on both sides proves the claim.

1. Assume the existence of a matrix A ∈ sl(2,C) with

exp A = B.

The two complex eigenvalues λi, i = 1,2, of A satisfy

0 = tr A = λ1 +λ2.

The case λ1 = λ2, i.e. 0 = λ1 = λ2 is excluded, because then exp A has an eigen-value e0 = 1. But B has the single eigenvalue −1.

Hence λ1 6= λ2. Then A is diagonalizable: A matrix S ∈ GL(2,C) exists with

S ·A ·S−1 =

(λ1 00 λ2

).

We obtain

S ·B ·S−1 = S · (exp A) ·S−1 = exp(S ·A ·S−1) =

(eλ1 00 eλ2

)Hence B is diagonalizable, a contradiction.

2. Assume the existence of a matrix A ∈M(2×2,R) with

exp A = B.

The real matrix A has two complex eigenvalues λi, i = 1,2, which are conjugateto each other.

On one hand, if both eigenvalues are real, then

λ1 = λ2 =: λ ∈ R

and exp A has an eigenvalue eλ > 0, while B has the single eigenvalue −1, acontradiction.On the other hand, if λ1 is not real, then

λ2 = λ1 6= λ1.

Hence A has two distinct eigenvalues. Therefore A is diagonalizable and also B,a contradiction, q.e.d.


Proposition 1.27 (Lie product formula). For any two matrices A,B ∈M(n×n,C)the exponential satisfies

exp(A+B) = limν→∞

(exp

Aν· exp

Bν

)ν

.

In the proof we will use for a power series f the standard notation

f = O(1/ν2)

iff for limν→∞

the quotient

f1/ν2 = ν

2 · f

remains bounded.

Proof. We conside the Taylor series

expAν= 1+

Aν+O(1/ν

2)

expBν= 1+

Bν+O(1/ν

2)

and

expAν· exp

Bν= 1+

Aν+

Bν+O(1/ν

2).

For large ν ∈ N we have ∥∥∥∥∥expAν· exp

Bν−1

∥∥∥∥∥< log 2.

Therefore the logarithm is well-defined:

log

(exp

Aν· exp

Bν

)= log

(1+

Aν+

Bν+O(1/ν

2)

)=

Aν+

Bν+O(1/ν

2).

Hence on one hand,

(exp log)

(exp

Aν· exp

Bν

)= exp

(Aν+

Bν+O(1/ν

2)

).

On the other hand, Proposition 1.23 implies

(exp log)

(exp

Aν· exp

Bν

)= exp

Aν· exp

Bν.


We get

expAν· exp

Bν= exp

(Aν+

Bν+O(1/ν

2)

)and(

expAν· exp

Bν

)ν

=

(exp

(Aν+

Bν+O(1/ν

2)

))ν

= exp(A+B+O(1/ν)).

Taking limν→∞

and using limν→∞

O(1/ν) = 0 and the continuity of the exponential provesthe claim

limν→∞

(exp

Aν· exp

Bν

)ν

= exp(A+B), q.e.d.

Chapter 2Fundamentals of Lie algebra theory

In this chapter the base field is either K= R or K= C.

2.1 Definitions and first examples

In general, in the associative algebra M(n×n,K) the product of two matrices is notcommutative:

AB 6= BA, A,B ∈M(n×n,K).

The commutator[A,B] := AB−BA ∈M(n×n,K)

is a measure for the degree of non-commutativity. The commutator depends K-linearlyon both matrices A and B. In addition, it satisfies the following rules:

1. Permutation of two matrices: [A,B]+ [B,A] = 0.

2. Cyclic permutation of three matrices: [A, [B,C]]+ [B, [C,A]]+ [C, [A,B]] = 0.

These properties make

gl(n,K) := (M(n×n,K), [−,−])

the prototype of a Lie algebra.

Definition 2.1 (Lie algebra).

1. A K-Lie algebra is a K-vector space L together with a K-bilinear map

[−,−] : L×L→ L (Lie bracket)

such that

• [x,x] = 0 for all x ∈ L

35

36 2 Fundamentals of Lie algebra theory

• [x, [y,z]]+ [y, [z,x]]+ [z, [x,y]] = 0 for all x,y,z ∈ L (Jacobi identity).

2. A morphism of K-Lie algebras is a K-linear map f : L1→ L2 between two K-Liealgebras satisfying

f ([x1,x2]) = [( f (x1), f (x2))],x1,x2 ∈ L1.

Note that the Lie bracket satisfies

[x,y]+ [y,x] = 0 f or all x,y ∈ L.

For the proof one computes [x+ y,x+ y] ∈ L.

We have just seen that the Lie algebra gl(n,K) derives from the associative al-gebra M(n× n,K). Actually, any finite dimensional K-Lie algebra derives from amatrix algebra, see Theorem ??. But the theory becomes more transparent whenconsidering Lie algebras and their morphisms as abstract mathematical objects.

A Lie algebra is a vector space with the Lie bracket as additional operator. Thisadditional algebraic structure resembles the multiplication within a group or a ring.The Lie algebra concept of the commutator is taken from group theory while theconcept of an ideal comes from ring theory.

Definition 2.2 (Basic algebraic concepts). Consider a K-Lie algebra L. One de-fines:

• A vector subspace M ⊂ L is a subalgebra of L iff [M,M]⊂M, i.e. iff M is closedwith respect to the Lie bracket of L.

• A vector subspace I ⊂ L is an ideal of L iff [L, I]⊂ I, i.e. if I is L-invariant.

• The normalizer of a subalgebra M ⊂ L is the subalgebra

NL(M) := x ∈ L : [x,M]⊂M ⊂ L,

i.e. the largest subalgebra which includes M as an ideal.

• The center of L is the ideal

Z(L) := x ∈ L : [x,L] = 0.

The center collects those elements which commute with all elements from L.

• The centralizer CL(Y ) of a subset Y ⊂ L is the largest subalgebra of L whichcommutes with Y

CL(Y ) := x ∈ L : [x,Y ] = 0

2.1 Definitions and first examples 37

• The derived algebra or commutator algebra of L is the subalgebra generated byall commutators

[L,L] := spanK[x,y] : x,y ∈ L.

Iff [L,L] = 0 then L is Abelian.

For a morphism f : L1→ L2 between to K-Lie algebras the kernel ker( f ) ∈ L1 isan ideal.

Apparently the derived algebra [L,L]⊂ L is an ideal.

Definition 2.3 (Specific matrix algebras).

1. For a finite dimensional K-vector space V we define the K-Lie algebra

gl(V ) := (End(V ), [−,−])

with[ f ,g] := f g−g f .

In particulargl(n,K) := gl(Kn).

Subalgebras of the Lie algebra gl(n,K) are named matrix algebras.

2. The subalgebra of strictly upper triangular matrices

n(n,K) := A = (ai j) ∈ gl(n,K) : ai j = 0 if i≥ j.

Each strictly upper triangular matrix has the form0 ∗. . .

0 0

3. The subalgebra of upper triangular matrices

t(n,K) := A = (ai j) ∈ gl(n,K) : ai j = 0 if i > j.

Each upper triangular matrix has the form∗ ∗. . .

0 ∗

4. The subalgebra of diagonal matrices

d(n,K) := A = (ai j) ∈ gl(n,K) : ai j = 0 if i 6= j.


Each diagonal matrix has the form∗ 0. . .

0 ∗

All three vector spaces of matrices are stable with respect to the commutator ofmatrices, hence they are subalgebras of gl(n,K).

Moreover for n≥ 2

n(n,K)( t(n,K)( gl(n,K) and d(n,K)( t(n,K).

We will see in Chapter 3.1 and Chapter 3.2 that n(n,K) and t(n,K) are the prototypeof the important classes of respectively nilpotent and solvable Lie algebras.

The fundamental tool for studying Lie algebras is the theory of their representa-tions. To represent an abstract Lie algebra L means to define a map from L to a Liealgebra of matrices. The concept of a representation has also many applications inphysics. We will study many examples in later chapters. The scope of the concept isnot restricted to Lie algebra theory.

Definition 2.4 (Representation of a Lie algebra). Consider a K-Lie algebra L.

1. A representation of L on a finite dimensional K-vector space V is a morphism ofLie algebras

ρ : L→ gl(V ) := (End(V ), [−,−]).

In particular,

ρ([x,y]) = [ρ(x),ρ(y)] = ρ(x)ρ(y)−ρ(y)ρ(x).

The vector space V is named L-module with respect to the multiplication

L×V →V,(x,v) 7→ x.v := ρ(x)(v),

which satisfies

[x,y].v = ρ([x,y])(v) = [ρ(x),ρ(y)](v) = ρ(x)(ρ(y)(v))−ρ(y)(ρ(x)(v)) =

x.(y.v)− y.(x.v).

The representation is faithful iff ρ is injective, i.e. ρ embeds L into a Lie algebraof matrices.

2.1 Definitions and first examples 39

2. A linear mapf : V −→W

between two L-modules is a morphism of L-modules if for all x ∈ L, v ∈V :

f (x.v) = x. f (v).

3. The adjoint representation of L is the map

ad : L→ gl(L), x 7→ ad x,

defined asad x : L→ L,y 7→ (ad x)(y) := [x,y].

Note. One often speaks about an L-module V without naming explicitly the rep-resentation

ρ : L→ gl(V ).

Lemma 2.5 (Adjoint representation of a Lie algebra). The adjoint representationis a representation, i.e. a morphism of Lie algebras.

Proof. For a Lie algebra L we have to show

ad [x,y] = [ad x,ad y] : L→ L.

Consider z ∈ L. On one hand,

ad[x,y](z) = [[x,y],z] =−[[y,z],x]− [[z,x],y] (Jacobi identity)

On the other hand

[ad x,ad y](z) := (ad x ad y−ad y ad x)(z) = [x, [y,z]]− [y, [x,z]].

Both results are equal because the Lie bracket is antisymmetric.

The adjoint representation maps any abstract Lie algebra to a matrix algebra.But in general the adjoint representation is not injective. The kernel of the adjointrepresentation of a Lie algebra L is the center Z(L). Nevertheless Theorem ?? willshow that any finite dimensional Lie algebra has a faithful representation, hence isa subalgebra of a Lie algebra of matrices.

The characteristic feature of a Lie algebra L is the Lie bracket. It refines theunderlying vector space of L. In order to study the Lie bracket one studies how eachsingle element x ∈ L acts on L as the endomorphism ad x.

This procedure is similar to the study of number fields Q ⊂ K ⊂ C. The multi-plication on K refines the underlying Q-vector space structure. And one studies the


multiplication by considering the Q-endomorphisms which result from the multipli-cation by all elements x ∈K. Norm and trace of these endomorphisms are importantparameters in algebraic number theory.

For all x∈ L the element ad x is not only an endomorphism of L, but also a deriva-tion of L: With respect to the Lie bracket it satisfies a rule similar to the Leibniz rulefor the derivation of the product of two functions.

Definition 2.6 (Derivation of a Lie algebra). Let L be a Lie algebra. A derivationof L is an endomorphism D : L→ L which satisfies the product rule

D([y,z]) = [D(y),z]+ [y,D(z)].

Lemma 2.7 (Adjoint representation and derivation). Consider a Lie algebra L.For every x ∈ L

ad x : L→ L

is a derivation of L.

Proof. Set D := ad x ∈ End(L). The Jacobi identity implies

D([y,z]) := (ad x)([y,z]) = [x, [y,z]] =−[y, [z,x]]− [z, [x,y]] = [y, [x,z]]+ [[x,y],z] =

= [y,(ad x)(z)]+ [(ad x)(y),z] = [y,D(z)]+ [D(y),z].

Derivations arising from the adjoint representation are named inner derivations,all other derivations outer derivations.

Lemma 2.8 (Algebra of derivations). Let L be a Lie algebra. The set of all deriva-tions of L

Der(L) := D ∈ End(L) : D derivation

is a subalgebra Der(L)⊂ gl(L).

Proof. Apparently Der(L)⊂ End(V ) is a subspace. In order to show that Der(L) iseven a subalgebra, we have to prove: If D1,D2 ∈ Der(L) then [D1,D2] ∈ Der(L).

[D1,D2]([x,y]) = (D1 D2)([x,y])− (D2 D1)([x,y]) =

= D1([D2(x),y]+ [x,D2(y)])−D2([D1(x),y]+ [x,D1(y)]) =

= [D1(D2(x)),y]+ [D2(x),D1(y)]+ [D1(x),D2(y)]+ [x,D1(D2(y)]

−[D2(D1(x)),y]− [D1(x),D2(y)]− [D2(x),D1(y)]− [x,D2(D1(y)] =

= [[D1,D2](x),y]+ [x, [D1,D2](y)].

2.2 Lie algebras of the classical groups 41

2.2 Lie algebras of the classical groups

An important class of Lie algebras are the Lie algebras of infinitesimal generatorsof the classical groups.

Definition 2.9 (1-parameter group and infinitesimal generator). For a matrix X ∈M(n×n,C)the differentiable group morphism

fX : (R,+)→ (GL(n,C), ·), t 7→ exp(t ·X),

with derivation (ddt

exp(t ·X)

)(0) = X (see Proposition 1.18).

is named the 1-parameter group of GL(n,C) with infinitesimal generator X .

Note: The general definition of a 1-parameter group of an arbitrary Lie group Gonly requires a continous group morphism

f : R→ G.

One can show: All 1-parameter groups of a Lie group G have the form

f (t) = exp(t ·X)

with an element X ∈ Lie G from the Lie algebra of G and the exponential map

exp : Lie G−→ G.

Definition 2.10 (Matrix group and 1-parameter subgroup).

1. A matrix group G is a closed subgroup

G⊂ GL(n,C).

Here GL(n,C) ⊂ Cn2is equipped with the subspace topology of the Euclidean

space.

2. Consider a matrix group G. For a matrix X ∈M(n×n,C) the group morphism

fX : R→ GL(n,C), t 7→ exp(t ·X),

is a 1-parameter subgroup of G with infinitesimal generator X iff for all t ∈ R

fX (t) ∈ G.


The term closed subgroup of G⊂GL(n,C) in Definition 2.10 refers to the topol-ogy of GL(n,C), i.e. a subgroup

G⊂ GL(n,C)

is closed iff for any sequence (Aν)ν∈N of matrices Aν ∈ H,ν ∈ N, which convergesin GL(n,C), i.e. with

A = limν→∞

Aν ∈ GL(n,C),

the limit belongs to G, i.e. A ∈ G.In the general context of a Lie group G one defines a Lie subgroup as a subgroup

which is also submanifold of G. One proves that any Lie subgroup of G is closedand a Lie group itself. Therefore Definition 2.10 requires a matrix group to be aclosed subgroup.

Proposition 2.11 (The classical groups and the Lie algebra of their infinitesimalgenerators). Consider a parameter r ∈N determining the dimension m = m(r) ∈Nof the underlying vector space Km, the matrix group

G := GL(m,K),

and the Lie algebraM := gl(m,K).

The classical groups and the Lie algebra of their infinitesimal generators are:

i) Series Ar,r ≥ 1,m := r + 1: The infinitesimal generators of all 1-parametersubgroups of the special linear group

SL(m,K) := g ∈ G : det g = 1 ⊂ G

form the subalgebra of traceless matrices

sl(m,K) := X ∈M : tr X = 0.

ii) Series Br,r ≥ 2,m := 2r + 1: The infinitesimal generators of all 1-parametersubgroups of the special orthogonal group

SO(m,K) := g ∈ G : g ·g> = 1,det g = 1

form the subalgebra of skew-symmetric matrices

so(m,K) := X ∈M : X +X> = 0.

iii) Series Cr,r ≥ 3,m = 2r: The infinitesimal generators of all 1-parameter sub-groups of the symplectic group

Sp(m,K) := g ∈ G : g> ·σ ·g = σ


with

σ :=(

0 1

−1 0

),σ−1 =

(0 −11 0

)=−σ ,1 ∈ GL(r,K) unit matrix,

form the subalgebra

sp(m,K) := X ∈M : X> ·σ +σ ·X = 0.

iv) Series Dr,r ≥ 4,m = 2r: The infinitesimal generators of all 1-parameter sub-groups of the special orthogonal group

SO(m,K) := g ∈ G : g ·g> = 1,det g = 1

form the subalgebra of skew-symmetric matrices

so(m,K) := X ∈M : X +X> = 0.

v) Special unitary group: The infinitesimal generators of all 1-parameter subgroupsof the special unitary group

SU(m) := g ∈ GL(m,C) : g ·g∗ = 1,det g = 1,g∗ := g> Hermitian con jugate,

span the real Lie algebra of traceless skew-Hermitian matrices

su(m) := X ∈ gl(m,C) : X +X∗ = 0, tr X = 0.

Note that any g ∈ Sp(m,K) has det g = 1, see [41].

Proof. We show for all cases that any infinitesimal generator of a 1-parameter sub-group of the matrix group belongs to the Lie algebra in question. Vice versa, Propo-sition 1.19 implies that any element of the Lie algebra is an infinitesimal generatorof a 1-parameter subgroup of the matrix group.

i) Taking the derivativeddt

on both sides of the equation

1 = det(exp(t ·X)) = exp(tr (t ·X))

gives0 = (tr X) · exp(tr (t ·X)).

Hence for t = 0:0 = tr X .

In the opposite direction: If tr X = 0 then

det etX = etr (tX) = et·(tr X) = e0 = 1.


Moreoversl(m,K) = ker[gl(m,K)

tr−→K]

is the kernel of the trace morphism to K considered as Abelian Lie algebra.Hence sl(m,K)⊂ gl(m,K) is even an ideal and a posteriori a subalgebra.

ii) and iv) Taking the derivative of

1= exp(t ·X) · exp(t ·X)> = exp(t ·X) · exp(t ·X>)

and using the product rule gives

0 = X · exp(t ·X) · exp(t ·X>)+ exp(t ·X) ·X> · exp(t ·X>).

Hence for t = 0:X +X> = 0.

In the opposite direction:X +X> = 0

impliesexp(tX) · exp(tX>) = exp(t(X +X>)) = e0 = 1

and tr X = 0 impliesdet(exp(tX)) = etr X = 1.

And for X ,Y ∈ o(n×n,K):

[X ,Y ]+ [X ,Y ]> = XY −Y X +(XY −Y X)> = XY −Y X +Y>X>−X>Y> =

X(Y +Y>)+(Y>+Y )X> = 0

using X =−X>.

iii) Taking the derivative of

σ = exp(t ·X)> ·σ · exp(t ·X) = exp(t ·X>) ·σ · exp(t ·X)

gives

0 = X> · exp(t ·X>) ·σ · exp(t ·X)+ exp(t ·X>) ·σ ·X · exp(t ·X).

Hence for t = 0:0 = X> ·σ +σ ·X .

In the opposite direction:0 = X>+σ ·X ·σ−1

implies(exp(tX))> ·σ · exp(tX) ·σ−1 = 1.

And for X ,Y ∈ sp(2n,K):


[X ,Y ]> ·σ +σ · [X ,Y ] = (Y>X>−X>Y>) ·σ +σ · (XY −Y X) = 0

after inserting X> ·σ ·Y−X> ·σ ·Y and Y> ·σ ·X−Y> ·σ ·X and using X> ·σ =−σ ·X .

v) Taking the derivative of

1= exp(t ·X) · exp(t ·X)∗

and using the product rule gives

0 = X · exp(t ·X) · exp(t ·X∗)+ exp(t ·X) ·X∗ · exp(t ·X∗).

Hence for t = 0X +X∗ = 0.

Moreover det(exp(tX)) = 1 for all t ∈K implies

tr X = 0.

In the opposite direction:X +X∗ = 0

implies[X ,X∗] =−[X ,X ] = 0

and

exp(tX) · (exp(tX))∗ = exp(tX) · exp(tX∗) = exp(t(X +X∗)) = 1.

Andtr X = 0

impliesdet(exp(tX)) = etr X = 1.

One checks that su(m) is a real Lie algebra. But su(m) is not a complex Lie algebra:

X ∈ su(2) ⇐⇒ X =

(ia b−b −ia

), a ∈ R, b ∈ C.

If X ∈ su(2) and X 6= 0 then iX /∈ su(2), q.e.d.

The reason for introducing the series in Proposition 2.11 and for the distinctionbetween the two series Br and Dr will become clear later in Chapter 7. The lowerbound for the parameter r ∈ N has been choosen to avoid duplicates or direct sumdecompositions. Otherwise we would have for the base field K = C and the corre-sponding Lie algebras:

A1 = B1 =C1, C2 = A1⊕A1, D3 = A3.


• Elements of the complex matrix groups of the series Br, Dr preserve the K-bilinearform on Km

(z,w) =m

∑j=1

z j ·w j.

• Elements of the complex matrix groups of the series Cr preserve the K-bilinearform on Km

(x,y) =r

∑i=1

xi · yr+i− xr+i · yi.

• Elements of the special unitary group SU(m) preserve the sesquilinear (Hermi-tian) form on Cm

(z,w) =m

∑j=1

z j ·w j.

The bilinear form (z,w) is C-linear with respect to the first component and C-antilinear with respect to the second component.

Over the complex numbers quadratic forms always transform to these normalforms by a change of basis. Therefore one can restrict the investigation to the latter.The situation changes over the real numbers. Here one has to consider differentnormal forms, depending on the signature of the quadratic form in question.

The examples from Proposition 2.11 illustrate the general fact that the infinitesi-mal generators of all 1-parameter groups of a matrix group form a real Lie algebra.

Proposition 2.12 (Infinitesimal generators of a matrix group). For a matrixgroup

G⊂ GL(n,C)

the set of infinitesimal generators of all 1-parameter subgroups of G

L := X ∈M(n×n,C) : fX (t) ∈ G for all t ∈ R

is a real subalgebra of gl(n,C)R.Here gl(n,C)R is the real Lie algebra which results from the complex Lie

algebra gl(n,C) by restricting scalars.

Proof. i) If X ∈ L and s ∈ R then for all t ∈ R also

exp(t · (sX)) = exp((s · t)X) ∈ G.

ii) If X , Y ∈ L then Proposition 1.27 implies

exp(t(X +Y )) = limν→∞

(exp

tXν· exp

tYν

)ν

.

The closedness of G⊂ GL(n,C) implies


exp(t(X +Y )) ∈ G.

iii) We first show that for all X ∈ L and all A ∈ G also AXA−1 ∈ L: Proposition 1.19implies for all t ∈ R

exp(t(AXA−1)) = exp(A(tX)A−1) = A · exp(tX) ·A−1 ∈ G.

For arbitrary X , Y ∈ L the map

f (t) := R−→ GL(n,C), t 7→ exp(tX) ·Y · exp(−tX),

is a 1-parameter group with values in G:

f (t + s) = exp((t + s)X) ·Y · exp((−t− s)X) =

exp(tX)(exp(sX) ·Y · exp(−sX))exp(−tX) = exp(tX) f (s)exp(−tX) =

= f (t) · f (s)

andexp(sX) ·Y · exp(−sX) ∈ G =⇒ exp(tX) f (s)exp(−tX) ∈ G.

The chain rule and the product rule from Proposition 1.7 imply:

ddt

f (t) =ddt(exp(tX) ·Y · exp(−tX)) = X · exp(tX) ·Y · exp(−tX)− exp(tX) ·Y ·X

Henced fdt(0) = XY −Y X = [X ,Y ].

q.e.d.

Definition 2.13 (Real form of a complex Lie algebra, complexification of a realLie algebra).

1. Consider a complex Lie algebra L. By restricting scalars from C to R the Liealgebra L can be considered a real Lie algebra LR.

2. Consider a real Lie algebra M. The complexification of M is the complex Liealgebra

M⊗RC

with Lie bracket

[m1⊗ z1,m2⊗ z2] := [m1,m2]⊗ (z1 · z2), m1,m2 ∈M, z1,z2 ∈ C,

3. A real form of the complex Lie algebra L is a real subalgebra M ⊂ LR such thatthe complex linear map


M⊗RC→ L,m⊗1 7→ m,m⊗ i 7→ i ·m,

is an isomorphism of complex Lie algebras.

Remark 2.14 (Real forms are not necessarily unique). The complex Lie algebra sl(2,C)has the non-isomorphic real forms su(2) and sl(2,R):

i) Real forms: Apparently, sl(2,R) is a real form of sl(2,C). The C-linear map

su(2)⊗RC−→ sl(2,C), X⊗1 7→ X , X⊗ i 7→ iX ,

has the inverse C-linear map

sl(2,C)−→ su(2)⊗RC, Z = X + iY 7→ (−iX)⊗ i+ iY ⊗1

with

X :=Z +Z∗

2, Y :=

Z−Z∗

2i.

ii) 2-dimensional subalgebra of sl(2,R): The real Lie algebra sl(2,R) containsthe 2-dimensional subalgebra

spanR <

(1 00 −1

),

(0 10 0

)> .

iii) No 2-dimensional subalgebra of su(2): Introducing the trace zero HermitianPauli matrices

σ1 :=(

0 11 0

),σ2 :=

(0 −ii 0

),σ3 :=

(1 00 −1

)we obtain the basis (i ·σ j) j=1,2,3 of the real Lie algebra su(2) of skew-Hermitiantrace zero matrices. The Pauli matrices have the non-zero commutator

[σ1,σ2] = 2i ·σ3,

and all further non-zero commutators result from cyclic permutation.

We denote the infinitesimal generators of the standard 1-parameter groups ofrotations around the coordinate axes of R3 by

X :=

0 0 00 0 −10 1 0

, Y :=

0 0 10 0 0−1 0 0

, Z :=

0 1 0−1 0 00 0 0

∈ so(3,R).

Then one has an isomorphism of real Lie algebras

f : su(2) '−→ so(3,R)

2.3 Topology of the classical groups 49

defined byf (i ·σ3) := 2Z, f (i ·σ2) := 2Y, f (i ·σ1) := 2X .

The Lie algebra so(3,R) is isomorphic to the Lie algebra vector product

Vect := (R3,×).

But for any two linear independent vectors x1,x2 ∈ Vect the product x1× x2 is or-thogonal to the plane generated by x1 and x2. Hence

Vect ' so(3,R)' su(2)

does not contain a 2-dimensional subalgebra, q.e.d.

2.3 Topology of the classical groups

The present sections illustrates some topological properties of the classical matrixgroups. We illustrate how the investigation of matrix groups can be reduced to thestudy of simply connected matrix groups. The latter can be studied by their Liealgebras. The proofs are special cases from the general Lie group theory.

Proposition 2.15 (Topology of SO(3,R),SU(2),SL(2,C)).

1. The matrix group SO(3,R) is connected, the matrix group O(3,R) has two con-nected components.

2. The matrix group SU(2) - as a differentiable manifold - is diffeomorphic tothe 3-sphere:

SU(2)' S3 := x ∈ R4 : ‖x‖= 1.

In particular, SU(2) is simply connected.

3. The matrix group SL(2,C) - as a differentiable manifold - is diffeomorphic to theproduct

(C2 \0)×C.

The product is homeomorphic to

S3×R3.

In particular, SL(2,C) is simply connected.

Proof. 1. We show that the matrix group SO(3,R) is path-connected: For a givenrotation matrix A we choose an orthonormal basis of R3 with the rotation axis asthird basis element. Then we may assume


A =

cos δ sin δ 0−sin δ cos δ 0

0 0 1

∈ SO(3,R),δ ∈ [0,2 ·π[.

The path

γ : [0,1]→ SO(3,R), t 7→

cos (t ·δ ) sin (t ·δ ) 0−sin (t ·δ ) cos (t ·δ ) 0

0 0 1

∈ SO(3,R)

is a continuous map and connects γ(0) = 1 with γ(1) = A in SO(3,R).

Each matrix A ∈ O(3,R) hasdet A =±1.

Hence O(3,R) has two connected components: SO(3,R), the connected compo-nent of the identity, and the residue class

SO(3,R) ·

1 0 00 1 00 0 −1

.

As a consequence, both Lie groups SO(3,R) and O(3,R) have the same Lie al-gebra

so(3,R) = o(3,R).

2. Consider a matrix

A =

(a zc w

)∈ SU(2).

Then

A−1 =

(w −z−c a

)and A∗ =

(a zc w

).

HenceSU(2) = A ∈ GL(2,C) : A ·A∗ = 1, det A = 1=(

w z−z w

): z, w ∈ C, |z|2 + |w|2 = 1

'

' S3 ⊂ C2 ' R4

is homeomorphic to the 3-dimensional unit sphere. The unit sphere S3 is com-pact, connected and simply connected. Simply connected means the vanishing ofthe fundamental group

π1(S3,∗) = 0

or equivalently: Any closed path in S3 is contractible in S3 to one point. Intu-itively, simple connectedness means that S3 has no “holes”.


The result π1(S3,∗) = 0 is a particular case of the theorem by Seifert and vanKampen, see [46, Kap. 5.3]: One decomposes S3 as the union of its northern andsouthern hemispheres, which intersect in a space homeomorphic to S2.

3. Set B := C2 \0. On one hand, projecting a matrix onto its last column definesthe differentiable map

p : SL(2,C)→ B, A =

(a zc w

)7→(

zw

).

On the other hand, one considers the open covering U = (U1,U2) of the base Bwith

U1 := (z,w) ∈ B : z 6= 0, U2 := (z,w) ∈ B : w 6= 0

One obtains a well-defined map

f : SL(2,C)→ B×C

A =

(a zc w

)7→

(p(A),(a−w · r−2) · z−1) if p(A) ∈U1

(p(A),(c+ z · r−2) ·w−1) if p(A) ∈U2

withr2 := |z|2 + |w|2.

Note: If p(A) ∈U1∩U2 then

(a−w · r−2) · z−1 = (c+ z · r−2) ·w−1.

The map f is a differentiable isomorphism satisfying pr1 f = p. We have thediffeomorphisms

B = C2 \0 ' S3× ]0,∞[' S3×R.

Because S3 is simply connected according to part 1), also B and eventually

SL(2,C)' B×C

are simply connected, q.e.d.

Several relations exists between the classical groups. These relations can be madeexplicit by differentiable group morphisms. Hence it is advantegeous not to studyeach group in isolation. Instead, one should focus on the relations between the clas-sical groups and study those properties, which the groups have in common. We givetwo examples in low dimension.

Remark 2.14 started from the two different real Lie algebras

sl(2,R) 6= su(2).


Nevertheless, both have the same complexification

sl(2,R)⊗RC' sl(2,C)' su(2)⊗RC.

We now show the identity of the two real Lie algebras

su(2)' so(3,R).

The corresponding matrix groups

SU(2) and SO(3,R)

are not equal. But Example 2.16 constructs a 2-fold covering projection

Φ : SU(2)−→ SO(3,R).

Due to Proposition 2.15 the group SU(2) is simply connected. The existence ofthe 2-fold covering projection Φ implies that SO(3,R) is not simply connected.

Example 2.16 (SU(2) as two-fold covering of SO(3,R)).

In order to obtain a suitable morphism

Φ : SU(2)→ SO(3,R)

we have to find out first: How does a complex matrix U ∈ SU(2) act on the 3-dimensionalreal space R3?

i) Vector space of Hermitian matrices: We define the real vector space of complexselfadjoint two-by-two matrices with trace zero

Herm0(2) := X ∈M(2×2,C) : X = X∗, tr X = 0.

The family of Pauli matrices (σ j) j=1,2,3 is a basis of Herm0(2). We define the map

α : R3→ Herm0(2),x = (x1,x2,x3) 7→ X :=3

∑j=1

x j ·σ j =

(x3 x1− i · x2

x1 + i · x2 −x3

).

On R3 we consider the Euclidean quadratic form

qE : R3→ R,x = (x1,x2,x3) 7→3

∑j=1

x2j .

Correspondingly, on the real vector space Herm0(2) we consider the quadratic form

qH : Herm0(2)→ R,X 7→ −det X ,

i.e.


qH(X) = a2 + |b |2 f or X =

(a bb −a

),a ∈ R,b ∈ C.

Then the mapα : (R3,qE)

∼−→ H := (Herm0(2),qH)

is an isometric isomorphism of Euclidean spaces , i.e. α is an isomorphism of realvector spaces with

qH(α(x)) = qE(x), x ∈ R3.

By means of the isometric isomorphism α we identify O(3,R), the group ofisometries of (R3,qE), with the group of isometries of H

O(H) := g ∈ GL(Herm0(2)) : qH(g(X)) = qH(X) f or all X ∈ Herm0(2).

Moreover, we identify the subgroup SO(3,R)⊂ O(3,R) with the subgroup

SO(H) := g ∈ O(H) : det g = 1 ⊂ O(H),

the connected component of the neutral element e ∈ O(H).

ii) Definition of Φ : We define the group morphism

Φ : SU(2)→ SO(H),B 7→ΦB,

by settingΦB : Herm0(2)→ Herm0(2), X 7→ B ·X ·B−1.

Note B ·X ·B−1 ∈ Herm0(2), because B−1 = B∗ and X∗ = X imply

(B ·X ·B−1)∗ = (B ·X ·B∗)∗ = B ·X ·B∗ = B ·X ·B−1.

We have ΦB ∈ O(H) because

−det(B ·X ·B−1) =−det X .

BecauseSO(H)' SO(3,R)

is connected due to Proposition 2.15, we even have

ΦB ∈ SO(H).

iii) Tangent map of Φ : We use from the theory of Lie groups without proof:

• The underlying vector space of the Lie algebra of a matrix group G is the tangentspace Lie G = TeG at the neutral element e ∈ G.

• For each differentiable homomorphism of matrix groups

Ψ : G1→ G2


the induced tangent map at the neutral element e ∈ G1 is a morphism

ψ := Lie Ψ := TeΨ : Lie G1→ Lie G2

of Lie algebras.

It is common, to interpret the tangent map of the differentiable map Φ at e ∈ G1 asthe linearization of Ψ at e.

We now determine the linearization of

Φ : SU(2)→ SO(H), B 7→ΦB,

as a Lie algebra morphism

φ : su(2)−→ so(H), A 7→ φA.

Here so(H)⊂ gl(Herm0(2)) is the subalgebra of the infinitesimal generators ofall 1-parameter subgroups of SO(H).

We compute for

B = exp A with A ∈ su(2), X ∈ Herm0(2) :

ΦB(X) = Φexp A(X) = exp(A) ·X · exp(−A) =

= (1+A+O(A2)) ·X · (1−A+O(A2)) =

= X +A ·X−X ·A+O(A2) = X +[A,X ]+O(A2).

As linearisation with respect to A ∈ su(2) we obtain

ϕA : H −→ H, X 7→ ϕA(X) = [A,X ].

Similarly to Lemma 2.5 one verifies that ϕ is indeed a morphism of Lie algebras.

To determine the value ofϕ : su(2)−→ so(H)

on the basis elements (i ·σ j) j=1,2,3 of su(2) we compute:

ϕiσ1 : H→ H

ϕiσ1(σ1) = 0,ϕiσ1(σ2) = i · [σ1,σ2] =−2 ·σ3

ϕiσ1(σ3) = i · [σ1,σ3] =−i · [σ3,σ1] = 2 ·σ2

With respect to the basis (σ j) j=1,2,3 of H we obtain

ϕiσ1 = 2 ·

0 0 00 0 10 −1 0

∈ so(H).


A similar evaluation of φ on the other two basis elements i ·σ2 and i ·σ3 gives

ϕiσ2 = 2 ·

0 0 −10 0 01 0 0

,ϕiσ3 = 2 ·

0 1 0−1 0 00 0 0

∈ so(H).

Hence the family (ϕi·σ j) j=1,2,3 is linearly independent in so(H). As a consequence,

ϕ : su(2) ∼−→ so(H)

is an isomorphism of Lie algebras.

iv) Surjectivity of Φ : The morphism

Φ : SU(2)→ SO(3,R)

is a local isomorphism at 1 ∈ SU(2) because its tangent map is bijective due topart iii). In particular, Φ is an open map. The image

Φ(SU(2))⊂ SO(3,R)

is compact because SU(2) is compact due to Proposition 2.15.As a consequence, the open and closed subset Φ(SU(2))⊂ SO(3,R) equals the

connected set SO(3,R), i.e. the map Φ is surjective.

v) Discrete kernel: We are left with calculating the kernel of Φ . We claim:

ker Φ = ±1 ⊂ SU(2).

For the proof consider an arbitrary but fixed matrix

B =

(z w−w z

)∈ SU(2)

with B ·X ·B−1 = X for all X ∈ Herm0(2). Then

B−1 = B∗ =(

z −ww z

).

Choosing for X successively the basis elements σ1,σ2 ∈ Herm0(2) we obtain

B ·σ1 ·B−1 = σ1 and B ·σ2 ·B−1 = σ2.

Comparing on both sides the components and using the determinant condition

1 = det B = |z|2 + |w|2

gives


1 = |z|2−|w|2, w = 0, and z2 = 1.

Hence z =±1. As a consequence

B =±1.

vi) Universal covering space: We use from Lie group theory without proof that asurjective morphism between Lie groups with discrete kernel is a covering projec-tion. The group SU(2) is simply connected according to Proposition 2.15. Hencethe map

Φ : SU(2)→ SO(3,R)

is the univeral covering projection of SO(3,R), and SU(2) is a double coverof SO(3,R).

vii) Fundamental group of SO(3,R): As a consequence of part vi) we have

π1(SO(3,R),∗) = Z2.

Nearly the same method as in Example 2.16 allows to compute the universal cov-ering space of the connected component of the neutral element of the Lorentz group.More specific, the covering projection Φ of SO(3,R) from Example 2.16 extends toa covering projection Ψ of the orthochronous Lorentz group, see Proposition 2.18:

Remark 2.17 (Minkowski space and Lorentz group). Figure 2.1 shows Minkowskispace with the embedded light cone. Points of Minkowski space are named events.Referring to the event e at the origin, the closure of the interior of the light cone,the set with qM(x) ≤ 0, splits into the disjoint union of e, the future and the pastcone of e. This splitting refers to the causal structure of the world: Events from thepast may have influenced e, while e may influence events in its future. All otherevents, qM(x)> 0, have no causal relation to e. They are the present of e.

The Lorentz group is the matrix group of isometries of the Minkowski space

M := (R4,qM)

with the quadratic form of signature (3,1)

qM : R4→ R,qM(x) :=−x02 + x1

2 + x22 + x3

2, x = (x0, ...,x3)>.

Here x0 denotes the time coordinate while x j, j = 1,2,3, are the space coordinates.Using these conventions Euclidean space embeds into Minkowski space in a naturalway. Minkowski space has the group of isometries:

O(3,1) := f ∈ GL(4,R) : qM( f (x)) = qM(x) f or all x ∈ R4.

If


Fig. 2.1 World model of Minkowski space

η :=

−1 0 0 00 1 0 00 0 1 00 0 0 1

then

B ∈ O(3,1) ⇐⇒ η = B> ·η ·B.

Hence for B ∈ O(3,1):

−1 = det η = det(B> ·η ·B) = det B> ·det η ·det B =−(det B)2

which impliesdet B =±1.

For B = (b jk)0≤ j,k≤3 ∈ O(3,1) the time like vector

e0 := (1,0,0,0)>

satisfies−1 = qM(e0) = qM(B · e0) =−b2

00 +b210 +b2

20 +b230.

Henceb2

00 = 1+b210 +b2

20 +b230 ≥ 1.


The subgroup

L↑+ := B = (b jk)0≤ j,k≤3 ∈ O(3,1) : det B = 1, b00 ≥ 1

is connected because L↑+ is closed and also open:

L↑+ = B = (b jk)0≤ j,k≤3 ∈ O(3,1) : det B > 0 and b00 > 0.

The subgroupL↑+ ⊂ O(3,1)

is the connected component of 1 ∈ O(3,1): It is named the proper orthochronousLorentz group. The elements from L↑+ keep the orientation of vectors and the signof their time component.

The other three connected components of O(3,1) are

L↑− := B = (b jk)0≤ j,k≤3 ∈ O(3,1) : det B =−1, b00 ≥ 1

L↓+ := B = (b jk)0≤ j,k≤3 ∈ O(3,1) : det B = 1, b00 ≤−1

L↓− := B = (b jk)0≤ j,k≤3 ∈ O(3,1) : det B =−1, b00 ≤−1

Proposition 2.18 (Universal covering space of the Lorentz group). The properorthochronous Lorentz group has the universal covering projection

Ψ : SL(2,C)→ L↑+

with a suitable group homomorphism Ψ of real matrix groups, which is a two-foldcovering projection. The following diagram commutes

SU(2) SO(3,R)

SL(2,C) O(3,1)

Φ

Ψ

with the canonical inclusions in the vertical direction.

Proof. See also [45, Anhang L.8].

i) Minkowski space as a vector space of matrices: Let


H := (Herm(2),qH)

denote the real vector space of Hermitian matrices

Herm(2) := X ∈M(2×2,C) : X = X∗

equipped with the real quadratic form

qH : Herm(2)→ R,X 7→ −det X ,

i.e.qH(X) =−(a ·d)+ |b|2

for

X =

(a bb d

),a,d ∈ R,b ∈ C.

Set σ0 := 1 ∈ Herm(2). Then the family (σ j) j=0,...,3 is a basis of the vectorspace Herm(2). The map

β : M→ H,x = (x0, ...,x3) 7→ X :=3

∑j=0

x j ·σ j =

(x0 + x3 x1− i · x2

x1 + i · x2 x0− x3

)is an isometric isomorphism, i.e. an isomorphism of vector spaces satisfying

qH(β (x)) = qM(x), x ∈ R4.

The isometry property is due to

qH(β (x)) =−det β (x) =−x20 + x2

1 + x22 + x2

3 = qM(x).

By means of the isometric isomorphism β we identify the group of isometriesof H

O(H) := g ∈ GL(Herm(2)) : qH(g(X)) = qH(X) f or all X ∈ Herm(2)

with O(3,1) and denote byL↑+(H)⊂ O(H)

the connected component of the neutral element idH ∈ O(H).

ii) Definition of Ψ : The map

Ψ : SL(2,C)→ O(H),B 7→ΨB,

withΨB : H→ H,X 7→ B ·X ·B∗,

is a well-defined morphism of real matrix groups. We have


Ψ(SL(2C))⊂ L↑+(H)

because SL(2,C) is connected due to Proposition 2.15.

iii) Tangent map of Ψ : The family (A j) j=1,...,6 with

A j :=

σ j if j = 1,2,3

i ·σ j−3 if j = 4,5,6

is a basis of the real vector space sl(2,C)R. Note that

su(2) = spanR < i ·σ j : j = 1,2,3 >

is a real subalgebra of sl(2,C)R, but

spanR < σ j : j = 1,2,3 >

is not a subalgebra of sl(2,C)R. Denote by

o(H) = so(H)⊂ gl(Herm(2))

the subalgebra of infinitesimal generators of all 1-parameter subgroups of O(H).And let

ψ := Lie Ψ : sl(2,C)R→ o(H), A 7→ ψA,

be the tangent map of Ψ at 1 ∈ SL(2,C). It is a morphism of real Lie algebras:

ψA(X) = A ·X +X ·A∗, A ∈ sl(2,C)R, X ∈ Herm(2).

One checks that indeedψ[A1, A2] = [ψA1 ,ψA2 ]

by using the commutator relation of the Pauli matrices and the Hermitian properties

σ∗j =−σ j, (i ·σ j)

∗ = i ·σ j for j = 1,2,3.

Explicit computation of the matrices representing

ψA j , j = 1, ...,6,

shows: The family (ψA j) j=1,...,6is linearly independent in the vector space End(Herm(2)).

iv) Surjectivity of Ψ : The map

Ψ : SL(2,C)−→ L↑+(H)

is open because its tangent map 1 ∈ SL(2,C) is a local isomorphism. The image


Ψ(SL(2,C))⊂ L↑+(H)

is also closed because

L↑+(H) =⋃

g∈L↑+(H)

g ·Ψ(SL(2,C))

represents the complement

L↑+(H)\Ψ(SL(2,C))

as a union of open subsets. Hence

Ψ : SL(2,C)→ L↑+(H)

is surjective.

v) Discrete kernel: The kernel is

ker Ψ = ±1 ⊂ SL(2,C) :

For the proof one checks the condition

ΨB(X) = X

for all B ⊂ SL(2,C) and the matrices X ∈ Herm(2) from the basis (σ j) j=0,...,3of Herm(2).

As a consequence of part i) - part v):

Ψ : SL(2,C)→ L↑+(H)

is the universal covering projection of the proper orthochronous Lorentz group. It isa two-fold covering projection. Because SL(2,C) is simply connected according toProposition 2.15, the orthochronous Lorentz group has the fundamental group

π1(L↑+,∗) = Z2, q.e.d.

For arbitrary dimension we have the following topological results, e.g. see [25,Chap. 10, §2] and [27, Chap. 17]:

Remark 2.19 (More examples from the classical groups).

1. General linear group, m≥ 1:The group GL(m,C) is connected.The real group GL(m,R) has two connected components. The component of theidentity is not simply connected for m≥ 2.


2. Series Ar,r ≥ 1,m = r+1:For K ∈ R,C the group SL(m,K) is connected. The groups SL(m,C) are sim-ply connected. One has

π1(SL(m,R),∗) =

Z m = 2Z/2 m≥ 3

3. Series Br,r ≥ 2,m = 2r, and series Dr,r ≥ 4,m = 2r+1:For K∈ R,C the group SO(m,K) is connected. The groups SO(m,R) are com-pact. One has

π1(SO(m,R),∗) =

Z m = 2Z/2 m≥ 3

For m≥ 3 the universal covering of SO(m,R) is Spin(m,R). One has

Spin(3,R) = SU(2),

cf. Example 2.16. Moreover

Spin(4,R) = SU(2)×SU(2).

4. Series Cr : r ≥ 3,m = 2r: For K ∈ R,C the symplectic groups Sp(m,K) areconnected. The groups Sp(m,C)) are simply connected, the real symplecticgroups satisfy

π1(Sp(m,R),∗) = Z.

5. Special unitary group, m≥ 1:The group SU(m) is compact, connected and simply connected.

No complex group from this list is compact, because any complex, connectedand compact Lie group is Abelian, see [27, Prop. 15.3.7].

Proposition 2.20 (Compact real form). Each complex Lie algebra from Proposition 2.11has a compact real form, i.e. the real form can be choosen as the Lie algebra of acompact matrix group:

1. General linear group, m≥ 1: The real Lie algebra

u(m) = X ∈ gl(m,C);X +X∗ = 0


is a compact real form of gl(m,C). It is the Lie algebra of infinitesimal genera-tors of the unitary group U(m) which is a compact matrix group.

2. Series Ar, m = r + 1: The real Lie algebra su(m) is a compact real formof sl(m,C). It is the Lie algebra of infinitesimal generators of the compact matrixgroup SU(m).

3. Series Br, m = 2r: The real Lie algebra so(m,R) is a compact real formof so(m,C). It is the Lie algebra of infinitesimal generators of the compact realmatrix group SO(m,R).

4. Series Cr, m = 2r: The real Lie algebra

sp(m) = X ∈ gl(m,R) : g>σ −σ ·g∗ = 0

is a compact real form of sp(m,C). It is the Lie algebra of infinitesimal genera-tors of the compact symplectic group

Sp(m) :=U(m)∩Sp(m,C).

Here U(m) denotes the unitary group

U(m) := g ∈ GL(m,C) : g ·g∗ = 1.

5. Series Dr, m = 2r + 1: The real Lie algebra so(m,R) is a compact real formof so(m,C). It is the Lie algebra of infinitesimal generators of the compact realmatrix group SO(m,R).

Proof. All groupsSU(m),Sp(m),SO(m,R)

are closed subgroups of a suitable unitary group U(n). Hence it suffices to provethe compactness of U(n): A matrix A ∈U(n) preserves the Euclidean metric of Cn,hence ‖U‖sup ≤ ‖U‖= 1, q.e.d.

Remark 2.21 (Exponential map of classical Lie groups). On one hand, in general theexponential map

exp : Lie G→ G

is not surjective for a connected classical group G: According to Example 1.26 thematrix

B =

(−1 b0 −1

)∈ SL(2,R),b ∈ R∗,

has not the form B = exp A with a matrix A ∈ sl(2,R).On the other hand, the exponential map


exp : Lie G→ G

is surjective for a connected matrix group G if Lie G is the compact form of acomplex Lie algebra, see [25, Chap. II, Prop. 6.10].

Chapter 3Nilpotent Lie algebras and solvable Lie algebras

If not stated otherwise, all Lie algebras and vector spaces in this chapter will beassumed finite dimensional over the base field K= C or K= R.

3.1 Engel’s theorem for nilpotent Lie algebras

Recall Definition 1.10: An endomorphism f ∈ End(V ) with V a vector space isnilpotent iff an index n ∈ N exists with f n = 0.

Definition 3.1 (Ad-nilpotency). Consider a Lie algebra L. An element x ∈ L is ad-nilpotent iff the induced endomorphism of L

ad x : L→ L,y 7→ [x,y],

is nilpotent.

Nilpotency and ad-nilpotency refer to two different structures: Nilpotency iter-ates the associative product, while ad-nilpotency iterates the Lie product. If the Liealgebra results from a matrix algebra both concepts are related: A Lie algebra ofnilpotent endomorphisms of a vector space acts “nilpotent” on itself by the adjointrepresentation.

Lemma 3.2 (Nilpotency implies ad-nilpotency). Consider a vector space V , a Liealgebra L⊂ gl(V ) and an element x ∈ L. If the endomorphism

x : V →V

is nilpotent, then also

65

66 3 Nilpotent Lie algebras and solvable Lie algebras

ad x : L→ L

is nilpotent, i.e. the element x ∈ L is ad-nilpotent.

Proof. The endomorphism x ∈ End(V ) acts on End(V ) by left composition andright composition

l : End(V )→ End(V ),y 7→ x y,

r : End(V )→ End(V ),y 7→ y x.

Thenad x = l− r

because for all y ∈ End(V )

(ad x)(y) = [x,y] = (l− r)(y).

Nilpotency of x implies that both actions are nilpotent. Hence lN = rN = 0 for asuitable exponent N ∈ N.

Both actions commute: For all y ∈ End(V )

[l,r](y) = x (y x)− (x y) x = 0.

Hence the binomial theorem implies

(ad x)2N = (l− r)2N =2N

∑ν=0

(2Nν

)lν · r2N−ν · (−1)2N−ν = 0

because each summand has at least one factor equal to zero.

An important relation between the associative algebra End(V ) and the Liealgebra gl(End V ) derives from the Jordan decomposition of endomorphisms ac-cording to Theorem 1.15.

Proposition 3.3 (Jordan decomposition of the adjoint representation). Consideran n-dimensional complex vector space V , an endomorphism f ∈ End(V ) and itsJordan decomposition

f = fs + fn.

Then the Jordan decomposition of

ad f ∈ L := gl(End V ),

defined asad f : End(V )−→ End(V ), g 7→ [ f ,g],

isad f = ad fs +ad fn ∈ L.

In particular, ad fs is semisimple, ad fn is nilpotent and [ad fs,ad fn] = 0.

3.1 Engel’s theorem for nilpotent Lie algebras 67

Proof. According to Lemma 3.2 the endomorphism ad fn is nilpotent. In order toshow that ad fs is semisimple we choose a base (v1, ...,vn) of V consisting of eigen-vectors of fs with corresponding eigenvalues λ1, ...,λn.

Let (Ei j)1≤i, j≤n denote the standard base of End(V ) relatively to (v1, ...,vn), i.e.

Ei j(vk) = δ jkvi.

For 1≤ i, j,k ≤ n:

((ad fs)Ei j)(vk) = [ fs,Ei j](vk) = fs(Ei j(vk))−Ei j( fs(vk)) = ( fs−λk)(Ei j(vk)) =

= ( fs−λk)(δ jkvi) = (λi−λ j)(δ jkvi) = (λi−λ j) ·Ei j(vk)

Hence(ad fs)(Ei j) = (λi−λ j) ·Ei j

and ad fs acts diagonally on the vector space End(V ) with eigenvalues

λi−λ j, 1≤ i, j ≤ n.

Becausead : gl(V )→ L

is a morphism of Lie algebras:

[ad fs,ad fn] = ad([ fs, fn]) = 0, q.e.d.

Using the adjoint representation we generalize the concept of a semisimple en-domorphism to elements of arbitrary complex Lie algebra.

Definition 3.4 (Ad-semisimple element of a complex Lie algebra). Consider acomplex Lie algebra L. An element x∈L is ad-semisimple iff the endomorphism ad x ∈ End(L)is semisimple.

It is a trivial fact that a nilpotent endomorphism of a vector space V 6= 0 hasan eigenvector with eigenvalue zero. Theorem 3.5 strongly generalizes this fact: Itproves the existence of a common eigenvector for a whole Lie algebra of nilpotentendomorphisms.

Theorem 3.5 (Annihilation of a common eigenvector). Consider a vector space V 6= 0and a matrix algebra L⊂ gl(V ) of endomorphisms.

If each endomorphism x ∈ L is nilpotent, then all x ∈ L annihilate a commoneigenvector, i.e. a nonzero vector v ∈V exists with

x(v) = 0 f or all x ∈ L.


Proof. The proof goes by induction on dim L, while the dimension of the finite-dimensional vector space V is arbitrary. As noted before the theorem is true fordim L = 1.

For the induction step assume dim L > 1 and the theorem true for all Lie algebrasof less dimension.

i) Existence of an ideal of codimension 1: By assumption all x ∈ L are nilpotentendomorphisms of V . Hence by Lemma 3.2 each action

ad x : L→ L,x ∈ L,

is nilpotent.

Define the setA := K ( L : K subalgebra

of all proper subalgebras of L. We have A 6= /0, because any 0-dimensional vectorsubspace of L is also a subalgebra.

If I ∈A is a subalgebra of maximal dimension with respect to A , then (ad x)(I)⊂ Ifor all x∈ I. Hence the Lie algebra I with dim I < dim L acts on the vector space L/I,and the action derives from the adjoint representation of L:

ad x : L/I→ L/K,y+ I 7→ [x,y]+K,x ∈ I.

Nilpotency of each ad x, x∈ I, implies the nilpotency of each ad x,x∈ I. The induc-tion hypothesis applied to the Lie algebra I and the vector space V := L/I providesa common eigenvector for all ad x,x ∈ I,

y+ I ∈ L/I

belonging to eigenvalue zero. In particular y /∈ I but [x,y] ∈ I for all x ∈ I. As aconsequence

y ∈ NL(I)\ I

i.e. I is properly contained in its normalizer. Due to the maximality of the dimensionof I we have NL(I) /∈A , hence

NL(I) = L,

i.e. I ∈ L is an ideal. It induces a canonical projection of Lie algebras

π : L→ L/I.

In case dim L/I≥ 2 the inverse image π−1(K)⊂L of a 1-dimensional subalgebra K ⊂ L/Iwere a proper Lie subalgebra of L, properly containing I, a contradiction to the max-imality of I. Hence dim L/I = 1.

ii) Constructing a common eigenvector: The construction from part i) implies thevector space decomposition


L = I +K · x0

with a suitable element x0 ∈ L\ I. By induction assumption the elements of I anni-hilate a common non-zero eigenvector, i.e.

W := w ∈V : x(w) = 0 f or all x ∈ I 6= 0.

In addition the subspace W ⊂V is stable with respect to the endomorphism x0 ∈ End(V ),i.e. for all w ∈W holds

x0(w) ∈W :

For all x ∈ Ix(x0(w)) = x0(x(w))− [x0,x](w) = 0.

Here the first summand vanishes because w ∈W and x ∈ I. The second summandvanishes because I ⊂ L is an ideal and therefore [x0,x] ∈ I. The restriction

x0|W : W →W

is a nilpotent endomorphism and has an eigenvector v0 ∈W with eigenvalue 0. Fromthe decomposition

L = I +K · x0

follows x(v0) = 0 for all x ∈ L. This finishes the induction step, q.e.d.

The fact, that any Lie algebra L⊂ gl(V ) of nilpotent endomorphisms annihilatesa common eigenvector, allows the simultaneous triagonalization of all endomor-phisms of L to strict upper triangular matrices.

Corollary 3.6 (Simultaneous strict triagonalization of nilpotent endomorphisms).Consider an n-dimensional K-vector space V and a matrix algebra L⊂ gl(V ) of en-domorphisms of V .

Then the following properties are equivalent:

1. Each endomorphism x ∈ L is nilpotent.

2. A flag (Vi)i=0,...,n of subspaces of V exists such that L(Vi)⊂Vi−1 for all i= 1, ...,n.

3. The Lie algebra L is isomorphic to a subalgebra of the Lie algebra of strictlyupper triangular matrices

n(n,K) =

0 ∗

. . .0 0

∈ gl(n,K)

.

In particular, any endomorphism x ∈ L has zero trace.


Proof. 1) =⇒ 2): The proof goes by induction on n = dim V . The implication isvalid for n = 0.

Assume n> 0 and part 2) valid for all vector spaces of less dimension. Set V0 := 0.According to Theorem 3.5 all elements from L annihilate a common eigenvector v1 ∈V .Consider the quotient vector space

W :=V/(K · v1)

with canonical projectionπ : V →W.

Any endomorphism x ∈ L ⊂ gl(V ) annihilates V1. Hence it induces an endomor-phism x : W →W such that the following diagram commutes

V V

W W

x

π

xπ

In particular, the endomorphism x∈ End(W ) is nilpotent. The induction assumptionapplied to W with

dim W < dim V

and L⊂ gl(W ) provides a flag (Wi)i=0,...,n−1 of subspaces of W with

x(Wi)⊂Wi−1 f or all i = 1, ...,n−1.

Now defineVi := π

−1(Wi−1), i = 1, ...,n.

Thenx(Wi)⊂Wi−1

impliesx(Vi)⊂Vi−1, i = 1, ...,n.

2) =⇒ 3) For the proof one extends successively a basis (v1) of V1 to a basis (v j) j=1,...,iof Vi, i = 1, ...,n.

3) =⇒ 1) The proof is obvious, q.e.d.

A Lie algebra is Abelian when the commutator of any two elements vanishes.Nilpotent Lie algebras are a first step to generalize this property: A Lie algebra isnamed nilpotent when a number N ∈ N exists such that all N-fold commutatorsvanish.

Definition 3.7 (Descending central series and nilpotent Lie algebra). Consider aLie algebra L.


1. The descending central series of L is the sequence (CiL)i∈N of ideals CiL ⊂ L,inductively defined as

C0L := L and Ci+1L := [L,CiL], i ∈ N.

2. The Lie algebra L is nilpotent iff an index i0 ∈ N exists with Ci0L = 0. Thesmallest index with this property is named the length of the descending centralseries.

3. An ideal I ⊂ L is nilpotent if I is nilpotent when considered as a Lie algebra withrespect to the restricted Lie bracket.

One easily verifies Ci+1L⊂CiL and each CiL⊂ L, i ∈ N, being an ideal.

Apparently, any subalgebra and any quotient algebra of a nilpotent Lie algebra L isnilpotent too: The respective descending central series arise from the descendingcentral series of L. Concerning the reverse implication consider Lemma 3.8.

The Lie algebra n(n,K) of strictly upper triangular matrices and all subalgebrasof n(n,K) are nilpotent Lie algebras.

Lemma 3.8 (Central extension of a nilpotent Lie algebra). Consider a Lie al-gebra L and an ideal I ⊂ L which is contained in the center, i.e. I ⊂ Z(L). If thequotient L/I is nilpotent then L is nilpotent too.

Proof. Set L′ := L/I and consider the canonical projection

π : L→ L′

with kernelker π = I.

The descending central series of L and L′ = π(L) relate as

Ci(π(L)) = π(CiL), i ∈ N.

If Ci0L′ = 0 then π(Ci0L) = 0, i.e.

Ci0L⊂ I ⊂ Z(L).

The center satisfies [L,Z(L)] = 0, hence

Ci0+1L = [L,Ci0L] = 0, q.e.d.


In Lemma 3.8 one may not drop the assumption that the ideal I ⊂ L belongs tothe center.

Example 3.9 (Counter example for nilpotent extension). The descending central se-ries of the Lie algebra

L := t(2,K) =

(∗ ∗0 ∗

)∈ gl(2,K)

is

C0L = [L,L] = n(2,K) =K ·(

0 10 0

)and

C1L = [L,n(2,K)] = n(2,K) =C0L 6= 0.

Hence L is not nilpotent. Moreover

I := n(2,K)⊂ L

is a nilpotent ideal, and the quotient of diagonal matrices

L/I ' d(2,K)

is also nilpotent.

We now prove Engel’s theorem. It is the main result about nilpotent Lie alge-bras. It characterizes the nilpotency of a Lie algebra by the ad-nilpotency of all itselements. Engel’s theorem follows as a corollary from Theorem 3.5.

Theorem 3.10 (Engel’s theorem for nilpotent Lie algebras). A Lie algebra L isnilpotent if and only if every element x ∈ L is ad-nilpotent.

Proof. i) Assume that every endomorphism

ad x : L→ L,x ∈ L,

is nilpotent. According to Corollary 3.6 the Lie algebra ad L is isomorphic to asubalgebra of

n(n,K),n = dim L,

of strictly upper triangular matrices, hence nilpotent. Due to Lemma 3.8 the com-mutative diagram

L ad L

L/Z(L)

ad

π∼


implies the nilpotency of L.

ii) Suppose L to be nilpotent. An index i0 ∈ N exists with Ci0L = 0. Hence forall x ∈ L

adi0(x) = 0,

i.e. ad x is nilpotent, q.e.d.

From a categorical point of view the concept of a short exact sequence is a usefultool to handle injective or surjective morphisms and respectively their kernels andcokernels.

Definition 3.11 (Exact sequence of Lie algebra morphisms).

1. A chain complex of Lie algebra morphisms is a sequence of Lie algebra mor-phisms

(Lifi−→ Li+1)i∈Z

such that for all i ∈ Z the composition fi+1 fi = 0, i.e.

im[Li−1fi−1−−→ Li]⊂ ker[Li

fi−→ Li+1].

2. A chain complex of Lie algebra morphisms (Lifi−→ Li+1)i∈Z is exact or an exact

sequence of Lie algebra morphisms if for all i ∈ Z

im[Li−1fi−1−−→ Li] = ker[Li

fi−→ Li+1].

3. A short exact sequence of Lie algebra morphisms is an exact sequence of theform

0→ L0f0−→ L1

f1−→ L2→ 0.

A short exact sequence

0→ L0f0−→ L1

f1−→ L2→ 0

expresses the following fact about two morphisms:

• f0 : L0→ L1 is injective,• f1 : L1→ L2 is surjective and• im f0 = ker f1, in particular L2 ' L1/L0.

Using the concept of exact sequences we restate the relation between the nilpo-tency of a Lie algebra L, an ideal of L and a quotient of L as follows:


Proposition 3.12 (Nilpotency and short exact sequences). Consider a short exactsequence of Lie algebra morphisms

0→ L0→ L1π−→ L2→ 0.

1. If the Lie algebra L1 is nilpotent, then also L0 and L2 are nilpotent.

2. If L2 is nilpotent and L0 ⊂ Z(L1), then also L1 is nilpotent.

For a Lie algebra L one obtains for any i ∈ N a short exact sequence

0→CiL/Ci+1L→ L/Ci+1L→ L/CiL→ 0

By definition of the descending central series, the Lie algebra on the left-handside is contained in the center of the Lie algebra in the middle, i.e.

CiL/Ci+1L⊂ Z(L/Ci+1L).

For a nilpotent Lie algebra L and minimal index i0 with Ci0L = 0 the exact sequence

0→Ci0−1L→ L→ L/Ci0−1L

presents L as a central extension of L/Ci0−1L. Successively one obtains L as a finitesequence of central extensions of nilpotent Lie algebras, starting with the AbelianLie algebra Z(L)' L/C1L.

Corollary 3.13 (Centralizer in nilpotent Lie algebras). Consider a nilpotent Liealgebra L and an ideal I ⊂ L, I 6= 0. Then

Z(L)∩ I 6= 0.

In particularCL(I) 6= 0.

Proof. According to Theorem 3.10 all endomorphisms

ad x : L→ L,x ∈ L,

are nilpotent. Because I ⊂ L is an ideal each restriction

(ad x)|I : I→ I,x ∈ L,

is well-defined and also nilpotent. According to Theorem 3.5, applied to ad L⊂ gl(I),a non-zero element x0 ∈ I exists with

[L,x0] = 0,

i.e. 0 6= x0 ∈ Z(L)∩ I, q.e.d.

3.2 Lie’s theorem for solvable Lie algebras 75

Proposition 3.14 (Normalizer in nilpotent Lie algebras). Any proper subalgebra M ( Lof a nilpotent Lie algebra L is properly contained in its normalizer M ( NL(M).

Proof. We consider the descending central series of L. Because M ( L = C0L westart with

M ( M+C0L.

Due to Ci0L = 0 for a suitable index i0 ∈ N we end with

M = M+Ci0L.

Let i < i0 be the largest index with

M ( M+CiL.

ThenM = M+Ci+1L.

Therefore

[M+CiL,M]⊂ [M,M]+ [CiL,M]⊂M+[CiL,L]⊂M+Ci+1L = M.

We obtainM ( M+CiL⊂ NL(M),q.e.d.

3.2 Lie’s theorem for solvable Lie algebras

Solvability generalizes nilpotency. Solvable Lie algebras relate to nilpotent Lie al-gebras like upper triangular matrices to strictly upper triangular matrices.

Definition 3.15 (Derived series and solvable Lie algebra). Consider a Lie algebra L.

1. The derived series of L is the sequence (DiL)i∈N inductively defined as

D0L := L and Di+1L := [DiL,DiL], i ∈ N.

2. The Lie algebra L is solvable iff an index i0 ∈N exists with Di0L = 0. The small-est index with this property is named the length of the derived series.

3. An ideal I ⊂ L is solvable iff I is solvable when considered as Lie algebra.

By induction on i∈N one easily shows that each DiL, i∈N, is an ideal in L. Notethat

D1L = [L,L]

is the derived or commutator algebra. Comparing the derived series with the lowercentral series one has DiL⊂CiL for all i ∈ N. Hence for Lie algebras:


Abelian =⇒ nil potent =⇒ solvable.

Lemma 3.16 (Solvability and short exact sequences). Consider an exact sequenceof Lie algebra morphisms

0→ L0→ L1π−→ L2→ 0.

If any two terms of the sequence are solvable then the third term is solvable too.

Proof. We prove that solvability of both L0 and L2 implies solvability of L1. AssumeDi0L2 = 0 and D j0L0 = 0. For all i ∈ N

DiL2 = Diπ(L1) = π(DiL1).

Hence Di0L2 = 0 implies Di0L1 ⊂ L0. Then

Di0+ j0L1 ⊂ D j0 L0 = 0.

The proof of the other direction uses the following relations between the derivedseries:

DiL0 ⊂ DiL1, π(DiL1) = DiL2, q.e.d.

Example 3.9 demonstrates that an analogous statement concerning the nilpotencyof the Lie algebra in the middle is not valid.

Corollary 3.17 (Solvable ideals). Consider a Lie algebra L. The sum I + J of twosolvable ideals I, J ⊂ L is solvable.

Proof. We apply Lemma 3.16: The quotient I/(I ∩ J) of the solvable Lie algebra Iis solvable too. By means of the canonical isomorphy

(I + J)/J ' I/(I∩ J)

the exact sequence0→ J→ I + J→ (I + J)/J→ 0

implies the solvability ofI + J, q.e.d.

Definition 3.18 (Radical of a Lie algebra). The unique solvable ideal of L whichis maximal with respect to all solvable ideals of L is denoted rad L, the radical of L.

To verify that the concept is well-defined take rad L as a solvable ideal not con-tained in any distinct solvable ideal. For an arbitrary solvable ideal I ⊂ L also


I + rad L

is solvable by Lemma 3.16. The inclusion

rad L⊂ I + rad L

implies by the maximality of rad L that

rad L = I + rad L.

Hence I ⊂ rad L.

The following Lemma 3.19 prepares the proof of Theorem 3.20 and thereforealso of Lie’s theorem about solvable Lie algebras.

Lemma 3.19 (Common triangularization). Consider a K-vector space V and a K-matrix algebra L⊂ gl(V ) of endomorphisms of V . Consider an ideal I ⊂ L. Assume:All endomorphisms

x : V →V, x ∈ I,

have a common eigenvector v ∈V . Then the eigenvector equation

x(v) = λ (x) · v, x ∈ I,

defines a linear functional λ : I→K such that

λ ([x,y]) = 0

for all x ∈ I,y ∈ L.

Proof. We choose an arbitrary but fixed y ∈ L.

i) Common triangularization on a subspace W ⊂ V : Let n ∈ N∗ be a maximalindex such that the family of iterates

B = (v,y(v),y2(v), ...,yn−1(v))

is linearly independent. Denote by W ⊂ V the n-dimensional vector space spannedby B. By definition W is stable with respect to y, i.e. y(W )⊂W .

We claim: The family

(Wi := span < v,y(v), ...,yi−1(v)>)i=0,...,n

is a flag of W , invariant with respect to all endomorphisms x : W →W, x ∈ I.

We prove x(Wi)⊂Wi by induction on i = 0, ...,n.For the induction step i 7→ i+1 we consider x ∈ I and decompose


x(yi(v)) = (x y)(yi−1(v)) = [x,y](yi−1(v))+(y x)(yi−1(v)).

The induction assumption applied to yi−1(v) ∈Wi proves: For the first summand

[x,y](yi−1(v)) ∈ I(Wi)⊂Wi ⊂Wi+1

because yi−1(v) ∈Wi and [x,y] ∈ I, and for the second summand

(y x)(yi−1(v)) = y(x(yi−1(v))) ∈ y(Wi)⊂Wi+1.

Hence x(yi(v)) ∈Wi+1.

With respect to the basis B of W each restricted endomorphism

x|W : W →W, x ∈ I,

is represented by an upper triangular matrix

x|W =

∗ ∗. . .

0 ∗

∈ t(n,K).

ii) Diagonal elements of the upper triangular matrices: We claim that all diagonalelements of x|W are equal to λ (x), i.e.

x|W =

λ (x) ∗. . .

0 λ (x)

, x ∈ I.

The proof of the equation

x(yi(v))≡ λ (x) · yi(v) mod Wi

goes by induction on i = 0, ...,n. The induction start i = 0 is the eigenvalue equation

x(v) = λ (x) ·v.

For the induction step i−1 7→ i consider the decomposition from above

x(yi(v)) = [x,y](yi−1(v))+(y x)(yi−1(v)).

For the first summand[x,y](yi−1(v)) ∈Wi.

For the second summand

(y x)(yi−1(v)) = y(x(yi−1(v)))≡ λ (x) · yi(v) mod Wi


because by induction assumption

x(yi−1(v))≡ λ (x) · yi−1(v) mod Wi−1

and by definition y(Wi−1)⊂Wi.

iii) Vanishing of the trace of a commutator: We have shown: All elements x ∈ Iact on the subspace W ⊂V as endomorphisms with

trace(x|W ) = n ·λ (x).

For all x ∈ I also [y,x] ∈ I because I ⊂ L is an ideal. The trace of a commutator oftwo endomorphisms vanishes. Hence for all x ∈ I

0 = tr([y|W,x|W ]) = n ·λ ([y,x])

which proves λ ([y,x]) = 0, q.e.d.

The present section deals with eigenvalues of certain endomorphisms. We needthe fact that a polynomial with coefficients from K splits completely into linearfactors over K. Therefore we assume that the underlying field K is algebraicallyclosed, i.e. we now consider complex Lie algebras.

Theorem 3.20 (Common eigenvector of a complex solvable matrix algebra).Consider a complex vector space V 6= 0 and a complex solvable matrix algebra L⊂ gl(V ).

Then L has a common eigenvector: A vector v ∈V,v 6= 0, and a linear functional

λ : V → C

exist such thatx(v) = λ (x) · v f or all x ∈ L.

The proof of this theorem will imitate the proof of Theorem 3.5.

Proof. Without restriction we may assume L 6= 0. Then the proof goes by induc-tion on dim L. The claim trivially holds for dim L = 0. For the induction step weassume dim L > 0 and suppose that the claim is true for all solvable Lie algebras ofsmaller dimension.

i) Construction of an ideal I ⊂ L of codimL(I) = 1: The derived series of L endswith Di0 = 0, hence it starts with D1L ( D0L, i.e. L/[L,L] 6= 0. Let

π : L→ L/[L,L]

be the canonical projection of Lie algebras. The Lie algebra L/[L,L] is Abelian,hence solvable. Therefore any arbitrary choosen vector subspace D ⊂ L/[L,L] ofcodimension 1 is even an ideal D⊂ L/[L,L]. Then I := π−1(D) is an ideal of L with


codimL(D) := dim L−dim D = 1.

The latter formula on the codimension is a general statement from the theory ofvector spaces:

dim I = dim π−1(D) = dim D+dim [L,L]

implies

codimL I = dim L−dim I = dim L−dim [L,L]−dim D =

= dim L/[L,L]−dim D = codimL/[L,L]D.

ii) Subspace of eigenvector candidates: We restrict the action of L on V to anaction of I. By induction assumption applied to I we get an element 0 6= v ∈V anda linear functional λ : I→ C with

x(v) = λ (x) ·v f or all x ∈ I.

We define the non-zero subspace of V

W := v ∈V : x(v) = λ (x) ·v f or all x ∈ I.

The subspace W ⊂V is stable under the action of L, i.e. for v ∈W and y ∈ L holdsy(v) ∈W : For arbitrary x ∈ I,v ∈W,y ∈ L we have

x(y(v)) = [x,y](v)+ y(x(v)) = [x,y](v)+λ (x) · y(v) = λ ([x,y]) · y(v)+λ (x) · y(v).

Here[x,y](v) = λ ([x,y]) · y(v)

because [L, I]⊂ I. Lemma 3.19, applied to I ⊂ gl(W ), gives λ ([x,y]) = 0. Hence

x(y(v)) = λ (x) · y(v)

and y(v) ∈W .

iii) 1-dimensional complement: According to the choice of the ideal I ⊂ L inpart i) an element x0 ∈ L exists with the vector space decomposition

L = I⊕C · x0.

By part ii) the subspace W is stable under the action of the restricted endomor-phism x0|W . Because the field C is algebraic closed an eigenvector v0 ∈W of x0with eigenvalue λ ′ exists:

x0(v0) = λ′ ·v0.

The linear functional λ : I → C extends to a linear functional λ : L → C bydefining λ (x0) := λ ′. Then


x(v0) = λ (x) ·v0 f or all x ∈ L, q.e.d.

A corollary of Theorem 3.20 is Lie’s theorem about the simultaneous triangular-ization of a complex solvable matrix algebra.

Theorem 3.21 (Lie’s theorem on complex solvable matrix algebras). Considera n-dimensional complex vector space V . Each solvable complex matrix algebraL ⊂ gl(V ) is isomorphic to a subalgebra of the Lie algebra of upper triangularmatrices

t(n,C) =

∗ ∗

. . .0 ∗

∈ gl(n,C)

.

Proof. Similar to the proof of Corollary 3.6 we construct by induction on dim V aflag (Vi)i=0,...,n of V , each Vi stable with respect to all endomorphisms

x : V →V, x ∈ L.

First we set V0 := 0. Then we choose a common eigenvector v1 ∈V satisfying

x(v) = λ (x) ·v

for all endomorphisms x ∈ L, see Theorem 3.20. Set

V1 := C ·v1.

Successively we extend the basis (v1) of V1 to a basis B = (v1, ...,vn) of V with

Vi = spanC < v1, ...,vi >, i = 0, ...,n.

With respect to the basis B all endomorphism x ∈ L are represented by upper trian-gular matrices, q.e.d.

Corollary 3.22 (Solvability and nilpotency). Consider a complex Lie algebra L.Then L is solvable iff its commutator algebra D1L = [L,L] is nilpotent.

Proof. The derived series of L relates to the lower central series of the Lie algebraDL as

DiL⊂Ci−1(DL)

for all i ≥ 1. The proof goes by induction on i ∈ N. The inclusion holds for i = 1.Induction step i 7→ i+1:

Di+1L := [DiL,DiL]⊂ [Ci−1(DL),Ci−1(DL)]⊂ [DL,Ci−1(DL)] =Ci(DL).

i) Assume that DL is nilpotent. The vanishing of Ci0(DL) for an index i0 ∈ Nimplies the vanishing of Di0+1L. Hence L is solvable.


ii) Assume that L solvable and dim L = n. Then also ad L ⊂ gl(L) is solvableaccording to Lemma 3.16. Theorem 3.21 implies

ad L⊂ t(n,C).

If x ∈ DL = [L,L] then

ad x ∈ [t(n,C), t(n,C)]⊂ n(n,C).

By Engel’s theorem, see Theorem 3.10, the Lie algebra DL is nilpotent, q.e.d.

Note that Corollary 3.22 is also valid for an R-Lie algebra L, because with respectto complexification

DiL⊗RC= Di(L⊗RC),

and similarly for the descending central series.

3.3 Semidirect product of Lie algebras

The semidirect product of two Lie algebras is a Lie algebra structure on the productvector space of the two Lie algebras. The most simple case of a semidirect productis the direct product. It is obtained by taking the Lie bracket in each componentseparately. But using certain derivations one can define a Lie bracket which mixesthe Lie brackets of the components. The semidirect product is an important tool toconstruct new Lie algebras, and also for splitting Lie algebras into factors.

Definition 3.23 (Semidirect product). Consider two Lie algebras I and M with Liebrackets respectively [−,−]I and [−,−]M , together with a morphism of Lie algebrasto the Lie algebra of derivations

θ : M→ Der(I).

The semidirect product of I and M with respect to θ is the Lie algebra

I oθ M := (L, [−,−])

with vector space L := I×M and bracket

[−,−] : L×L→ L

[(i1,m1),(i2,m2)] := ([[i1, i2]I +θ(m1)i2−θ(m2)i1, [m1,m2]M)

for i1, i2 ∈ I,m1,m2 ∈M.

3.3 Semidirect product of Lie algebras 83

Proposition 3.24 (Semidirect product). Consider two Lie algebras I and M, and amorphism of Lie algebras

θ : M −→ Der(I).

1. The semidirect productI oθ M

is a Lie algebra.

2. The semidirect product fits into the exact sequence of Lie algebras

0−→ Ij−→ I oθ M π−→M −→ 0

with j(i) := (i,0), π(i,m) := m.

3. The exact sequence has a section against π , i.e. a morphism of Lie algebras exists

s : M s−→ I oθ M

satisfyingπ s = idM.

4. We haveI ' j(I)⊂ I oθ M

an ideal, andM ' s(M)⊂ I oθ M

a subalgebra.

Proof. i) In order to verify for

x = (x1,x2),y = (y1,y2),z = (z1,z2) ∈ I×M

the Jacobi identity in the form

[[x,y],z]+ [[y,z],x]+ [[z,x],y] = 0

we distinguish four cases:

• If x,y,z ∈ I then the claim follows from the Jacobi identity of I.

• If x,y ∈ I and z ∈M then

[x,y] = ([x1,y1],0); [[x,y],z] = ([[x1,y1],0]−θ(z2)([x1,y1]),0)= (−θ(z2)([x1,y1]),0)

[y,z] = ([y1,z1]−θ(z2)(y1),0)= ([−θ(z2)(y1),0), [[y,z],x] = (−[θ(z2)(y1),x1],0)

[z,x] = ([z1,x1]+θ(z2)(x1),0) = (θ(z2)(x1),0), [[z,x],y] = ([θ(z2)(x1),y1],0)

Because I is a Lie algebra, the claim reduces to


θ(z2)([x1,y1])+ [θ(z2)(y1),x1])− [θ(z2)(x1),y1]) = 0.

The latter equation holds because θ(z2) is a derivation of I.

• If x ∈ I and y,z ∈M then

[x,y] = (−θ(y2)(x1),0); [[x,y],z] = (θ(z2)(θ(y2)(x1)),0)

[y,z] = (0, [y2,z2]); [[y,z],x] = (θ([y2,z2])(x1),0)

[z,x] = (θ(z2)(x1),0); [[z,x],y] = (−θ(y2)(θ(z2)(x1)),0)

The claim reduces to

θ(z2)(θ(y2)(x1))+θ([y2,z2])(x1)−θ(y2)(θ(z2)(x1)) = 0.

The latter equation holds because θ is a morphism of Lie algebras.

• If x,y,z ∈M then the claim follows from the Jacobi identity of M, q.e.d.

ii) Exactness of the sequence is obvious: j is injective, π is surjective, and

im j = j(I) = ker π.

iii) The maps : M −→ I oθ M, m 7→ (0,m),

is a morphism of Lie algebras, satisfying

π(s(m)) = π((m,0)) = m.

iv) The kernel of a morphism of Lie algebras is an ideal:

ker π = im j = j(I)' I.

The image of a morphism of Lie algebras is a subalgebra. Moreover,

π s = idM

implies the injectivity of s. Hence

s(M)'M

is a subalgebra of I oθ M.

Note. The semidirect product I oθ M is a direct product, i.e.

I oθ M ' I×M


iff θ = 0. And the latter equation is equivalent to

s(M)⊂ I oθ M

being an ideal.

A corner stone of quantum mechanics is the canonical commutation relation

[P,Q] =−ih ·1

with P the momentum operator, Q the position operator, the unit operator 1, andPlanck’s constant

h =h

2π= 1.054×10−27erg · sec.

This relation has been introduced by Born and Jordan and termed strict quantumcondition (German: verscharfte Quantenbedingung), see [3, Equation (38)].

Proposition 3.25 (Heisenberg algebra). The set

heis(n) :=

0 p c

0 0n q0 0 0

⊂ n(n+2,R) : p = (p1, ..., pn) ∈ Rn,q = (q1, ...,qn)> ∈ Rn,c ∈ R

is a real n+2-dimensional subalgebra, named Heisenberg algebra of n-dimensionalquantum mechanics. In particular: heis(1) = n(3,R). A basis of heis(n)

(P1, ...,Pn,Q1, ...,Qn, I)

is defined as

• Pj = E1,1+ j, the matrix with the only non-zero entry p j = 1,• Q j = E1+ j,n+2, the matrix with the only non-zero entry q j = 1, and• I = E1,n+2, the matrix with the only non-zero entry c = 1

has the commutators

[Pj,Qk] = δ jk · I and [Pj, I] = [Q j, I] = [Pj,Pk] = [Q j,Qk] = 0.

Proof. Similarly to the case n = 1 one checks that heis(n) ⊂ n(n+ 2,R) is closedwith respect to the Lie bracket.

The commutator relations follow from the commutator formula

[Ei, j,Es,t ] = δ j,s ·Ei,t −δt,i ·Es, j, q.e.d.

The Heisenberg algebra is nilpotent, being a subalgebra of the nilpotent Lie al-gebra n(n+2,R).


The Heisenberg algebra captures the kinematics of an n-dimensional quantummechanical problem. For a complete description which also covers the dynamics ofthe problem one needs a further operator: The Hamiltonian H of the problem andits relation to the kinematical operators. How to extend the Heisenberg algebra toinclude also the Hamiltonian H?

The solution is to construct the dynamical Lie algebra of quantum mechanics asthe semidirect product of the Heisenberg algebra and the 1-dimensional Lie alge-bra which corresponds to the Hamiltonian. For the case of 1-dimensional quantummechanics see also [27, Example 5.1.19].

Definition 3.26 (Dynamical Lie algebra of quantum mechanics). For n ∈ N con-sider the Heisenberg algebra heis(n) of n-dimensional quantum mechanics with vec-tor space basis

(P1, ...,Pn,Q1, ...,Qn, I),

the 1-dimensional Abelian Lie algebra R with vector space basis

(H := 1),

and the morphism of Lie algebras

θ : R−→ Der(heis(n))

with

θ(H)(Pj) := Q j, θ(H)(Q j) :=−Pj, j = 1, ...,n, and θ(H)(I) := 0.

The semidirect productquant(n) := heis(n)oθ R

is the dynamical Lie algebra of n-dimensional quantum mechanics. It fits into theshort exact sequence

0−→ heis(n)j−→ quant(n) π−→ R−→ 0.

Lemma 3.27 (Dynamical Lie algebra of quantum mechanics). The dynamicalLie algebra of n-dimensional quantum mechanics quant(n) is a well-defined realLie algebra. Identifying heis(n) with the ideal

j(heis(n))⊂ heis(n)oθ R

and R with the subalgebra

s(R) := 0×R⊂ heis(n)oθ R


the distinguished commutators of quant(n) are

[H,Pj] = Q j, [H,Q j] =−Pj, j = 1, ...,n, [H, I] = 0.

Proof. i) We have to show that θ maps to derivations. Using the shorthand D := θ(H):

• Left-hand side:D([Pj,Pk)]) = D(0) = 0

Right-hand side:

[D(Pj),Pk]+ [Pj,D(Pk)] = [Q j,Pk]+ [Pj,Qk] =−δ jkE +δ jkE = 0.

• Left-hand side:D([Q j,Qk)]) = D(0) = 0

Right-hand side:

[D(Q j),Qk]+ [Q j,D(Qk)] = [−Pj,Qk]+ [Q j,−Pk] =−δ jkE +δ jkE = 0.

• Left-hand side:D([Pj,Qk)]) = D(δ jkE) = 0

Right-hand side:

[D(Pj),Qk]+ [Pj,D(Qk)] = [Q j,Qk]+ [Pj,−Pk] = 0.

• Left-hand side:D([I,Pj]) = D(0) = 0

Right-hand side:

[D(I),Pj]+ [I,D(Pj)] = 0+[I,Q j] = 0

• Left-hand side:D([I,Q j]) = D(0) = 0

Right-hand side:

[D(I),Q j]+ [I,D(Q j)] = 0− [I,Pj] = 0.

ii) The mapθ : R−→ Der(heis(n))

is a morphism of Lie algebras: Because R is 1-dimensional and Abelian, the claimfollows from [D,D] = 0.

iii) For j = 1, ...,n the distinguished commutators follow from

[H,Pj] = [(0,H),(Pj,0)] = ([0,Pj]+D(Pj), [H,0]) = Q j,


[H,Q j] = [(0,H),(Q j,0)] = ([0,Q j]+D(Q j), [H,0]) =−Pj,

[H, I] = [(0,H),(I,0)] = (D(I),0) = (0,0), [H, I] = 0, q.e.d.

Proposition 3.28 (Solvability of quant(n)). The dynamical Lie algebra of quantummechanics

quant(n) := heis(n)oθ R

is solvable, but not nilpotent. Its derived algebra is

D1quant(n) = heis(n).

Proof. The descending central series of L := quant(n) is

C1L := spanR < [H,Pj], [H,Q j], [H, I] : j = 1, ...,n >+ C1(heis(n)) =

= spanR < Pj,Q j, I >= heis(n)

andC2L = [L,heis(n)] =C1L.

Hence for all i ∈ NCiL 6= 0,

which shows that quant(n) is not nilpotent. Corollary 3.22 and the subsequent noteimply that L is solvable, q.e.d.

Remark 3.29 (Heisenberg picture of quantum mechanics).

1. Like classical mechanics also quantum mechanics distinguishes between statesand observables to describe a physical system, cf. [40]. Pure states, as themost simple states, are represented by the 1-dimensional subspaces of a com-plex Hilbert space Hilb with a Hermitian product < −,− >, and the observ-able are self-adjoint operators on Hilb. The expectation value of measuring theobservable Ω when the system has been prepared in state φ is

< φ |Ω |φ >:=< φ ,Ω(φ)>=< Ω∗φ ,φ > .

2. The temporal development of the system, i.e. its dynamics, is governed by theHamiltonian H of the system according to the Schrodinger equation, the ordinarydifferential equation

1i

ddt

φ(t)+H(φ(t)) = 0, normalizsation: h = 1.

The Hamiltonian H is the observable of the total energy of the system. TheSchrodinger picture considers the temporal development of the states and fixes


the observables. For conservative systems - and only these will be considered inthe sequel - the Hamiltonian has no explicit time dependency.

3. The Heisenberg picture takes the opposite approach: It fixes the states and con-siders the temporal development of the observables. This approach resembles theHamiltonian approach of classical mechanics using canonical coordinates. In theHeisenberg picture, the temporal development of an observable Ω is governedby the commutator equation

1i

...Ω(t) = [H,Ω(t)]+

∂

∂ tΩ(t).

Here the partial derivation∂

∂ tΩ vanishes in case Ω does not explicitly depend

on the time t. In particular, for a set of canonical observables P j, Q j, j = 1, ...,n:

1i

...Q j(t) = [H,Q j(t)] =−P(t)

1i

...P j(t) = [H,P j(t)] = Q j(t).

A simple example of a 1-dimensional quantum system is the 1-dimensional os-cillator. It has the Hamiltonian

H =1

2mP2 +

mω2

2Q2

with respect to the momentum observable P and the position observable Q. Thereal number ω denotes the oscillator frequency, the real number m denotes theoscillator mass.

Note. There is no finite dimensional, i.e. m-dimensional dimensional representation ρ

of heis(n), m 6= 0, with

ρ([Pj,Q j]) = ρ(I) = λ · (−i ·1m),λ 6= 0,

becausetr ρ([P,Q]) = 0, but tr(λ · (−i ·1m) 6= 0.

The main result on the representation theory of self-adjoint operators P and Q satis-fying the Born-Jordan commutation rules

[P,Q] =−ih ·1

is the Stone-von Neumann theorem, see [21, Theor. 14.8 and Chap. 9.8].

The mathematical formalism of quantum mechanics is well understood: The the-ory of self-adjoint operators in a Hilbert space forms part of the domain of functional


analysis. But the interpretation of the physical content and even more the philo-sophical implications of quantum mechanics are still debated. Its first elaboratedinterpretation was the Copenhagen Interpretation of Quantum Mechanics, see [24].

Chapter 4Killing form and semisimple Lie algebras

All vector spaces are assumed finite-dimensional if not stated otherwise. The com-position f2 f1 of two endomorphisms will be denoted as product f2 · f1 or also f2 f1.

4.1 Traces of endomorphisms

The present section introduces a new tool for the study of Lie algebras: Importantproperties of a Lie algebra L express themselves by a bilinear form, which derivesfrom the trace of the endomorphisms of a suitable representation of L.

Lemma 4.1 (Simple properties of the trace). Consider a vector space V and en-domorphisms x,y,z ∈ End(V ).

1. For nilpotent x holdstr x = 0.

2. With respect to two endomorphisms the trace is symmetric, i.e.

tr (xy) = tr (yx).

In particular, tr[x,y] = 0.

3. With respect to cyclic permutation the trace is invariant, i.e.

tr (xyz) = tr (yzx)

4. With respect to the commutator the trace is “associative”

tr ([x,y]z) = tr (x[y,z]).

91

92 4 Killing form and semisimple Lie algebras

Proof. 1) The endomorphism x can be represented by a strictly upper triangularmatrix, e.g. due to Corollary 3.6.

2)With respect to matrix representations of the endomorphisms note

tr (xy) = ∑i, j

xi jy ji = ∑i, j

y jixi j = tr (yx)

3) According to part 2) we have

tr(xyz) = tr(x(yz)) = tr((yz)x) = tr(yzx).

4) We have[x,y]z = xyz− yxz and x[y,z] = xyz− xzy.

The ordering yxz results from the ordering xzy by cyclic permutation. Hence part 3)implies

tr(yxz) = tr(yxz)

which proves the claim.

Definition 4.2 (Killing form). Let L be a K-Lie algebra.

1. The trace form of a representation

ρ : L→ gl(V )

on a K-vector space V is the symmetric bilinear map

β : L×L→K,β (x,y) := tr (ρ(x)ρ(y)).

2. The Killing form of L

κ : L×L→K,κ(x,y) := tr (ad(x)ad(y))

is the trace form of the adjoint representation ad : L→ gl(L),

The following Lemma 4.3 prepares the proof of Theorem 4.4.

Lemma 4.3 (Trace check for nilpotency). Consider an n-dimensional complexvector space V , two subspaces A ⊂ B ⊂ End(V ) and the subspace of endomor-phisms

M := x ∈ End(V ) : [x,B]⊂ A.

An arbitrary element x ∈M is nilpotent, if it satisfies the property:

For all y ∈M holds tr(xy) = 0.

4.1 Traces of endomorphisms 93

Proof. i) Jordan decomposition for endomorphisms of V: Consider an arbitrary butfixed endomorphism

x : V →V

which satisfies the assumption [x,B]⊂ A. According to Theorem 1.15 a unique Jor-dan decomposition exists

x = xs + xn.

For a suitable basis B of V the semisimple part xs is represented by a diagonalmatrix

xs = diag(λ1, ...,λn)

and the nilpotent part xn by a strictly upper triangular matrix. Denote by

E := spanQ < λ1, ...,λn >⊂ C

the Q-vector subspace generated by the eigenvalues of xs. In order to show

xs = 0

we will show E = 0 or E∗ = 0 for the dual space of Q-linear functionals

f : E→Q.

ii) Jordan decomposition for endomorphisms of End(V ): According to Proposi-tion 3.3 the endomorphism

ad x : End(V )−→ End(V )

has the Jordan decomposition

ad x = ad xs +ad xn.

Therefore a polynomialps(T ) ∈ C[T ], ps(0) = 0,

exists withad xs = ps(ad x).

Consider a Q-linear functional f : E→Q. The endomorphism

y : V →V

defined with respect to B by the matrix

y = diag( f (λ1), ..., f (λn)).

is semisimple by construction.According to the proof of Proposition 3.3 both ad xs and ad y act on End(V) as

diagonal matrices with respect to the standard basis (Ei j)1≤i, j≤n of End(V ) corre-


sponding to B:

(ad xs)(Ei j) = (λi−λ j) ·Ei j and (ad y)(Ei j) = ( f (λi)− f (λ j)) ·Ei j.

Choose an interpolation polynomial q(T ) ∈ C[T ] with supporting points

(λi−λ j, f (λi)− f (λ j))1≤i, j≤n.

By construction q(0) = 0 and

ad y = q(ad xs) : End(V )−→ End(V ),

because the latter two map each Ei j to ( f (λi)− f (λ j)) ·Ei j. Therefore,

ad y = (q ps)(ad x)

and also the polynomial (q ps)(T ) ∈ C[T ] satisfies (q ps)(0) = 0.

From

[x,B] = (ad x)(B)⊂ A

follows[y,B] = (ad y)(B)⊂ A,

i.e. y ∈M.

iii) Trace computation: The product

xn · y ∈ End(V )

is nilpotent, because xn is nilpotent and y is a diagonal matrix. Hence the assumption

0 = tr (xy) = tr (xs · y)+ tr(xn · y) = tr (xs · y) =n

∑i=1

λi · f (λi) ∈ E ⊂ C

with λi ∈ E, f (λi) ∈Q, i = 1, ...,n, implies

0 = f

(n

∑i=1

λi · f (λi)

)=

n

∑i=1

f (λi)2 ∈Q.

Therefore, all f (λi), i = 1, ...,n, vanish. Hence f = 0 because the family (λi)i=1,...,nspans E. As a consequence xs = 0, q.e.d.

Theorem 4.4 (Cartan’s trace criterion for solvability). Consider a complex vec-tor space V and a subalgebra L⊂ gl(V ). The following properties are equivalent:

• L is solvable

4.1 Traces of endomorphisms 95

• The trace formβ : L×L→ C,β (x,y) := tr (xy),

satisfies β ([L,L],L) = 0.

Proof. 1) Assume β ([L,L],L) = 0.

i) Reduction to nilpotent endomorphisms: According to Corollary 3.22 it sufficesto prove the nilpotency of the derived algebra DL = [L,L]. According to Engel’stheorem, see Theorem 3.10, therefore it suffices to prove: Every element x ∈ [L,L]is a ad-nilpotent. We prove that all endomorphisms

x : V →V,x ∈ [L,L]⊂ gl(V ),

are nilpotent, and then apply Lemma 3.2.

ii) Trace check for nilpotency: Consider a fixed but arbitrary element

x0 ∈ [L,L]⊂ gl(V ).

To show nilpotency of x0 we apply the trace check from Lemma 4.3 with A := [L,L], B := Land

M := x ∈ End(V ) : [x,L]⊂ [L,L].

Apparently x0 ∈ [L,L]⊂ L⊂M. It remains to show for all endomorphisms y ∈M:

tr (xy) = 0.

Because [L,L] has generators [u,v],u,v ∈ L, we may assume x0 = [u,v]. Accordingto Lemma 4.1, part iv)

tr (x0y) = tr ([u,v]y) = tr (u[v,y]) = tr ([v,y]u).

But [v,y] ∈ [L,L] because y ∈M. By assumption β ([L,L],L) = 0, which proves

tr (x0y) = 0.

Lemma 4.3 implies that x0 ∈ gl(V ) is nilpotent.

2) Assume L solvable.

According to Lie’s theorem, see Theorem 3.21, the Lie algebra L is isomorphicto a subalgebra of upper triangular matrices:

L⊂ t(n,C) and [L,L]⊂ n(n,C), n = dim V.

Denote by(Vi)i=0,...,n, n = dim V,


a corresponding flag of L-invariant subspaces of V . For x ∈ L, [u,v] ∈ [L,L],

[u,v](Vi)⊂Vi−1

andx[u,v](Vi)⊂ x(Vi−1)⊂Vi−1, i = 1, ...,n.

Therefore x · [u,v] is nilpotent and

0 = tr (x[u,v]) = tr ([u,v]x).

Note. Theorem 4.4 is valid also in the real case.

4.2 Fundamentals of semisimple Lie algebras

Definition 4.5 (Simple and semisimple Lie algebras). Consider a Lie algebra L.

1. L is semisimple iff L has no Abelian ideal I 6= 0.

2. L is simple iff L is not Abelian and L has no ideal other than 0 and L.

These concepts apply also to an ideal I ⊂ L when the ideal is considered a Liealgebra.

Note: The trivial Lie algebra L= 0 is semisimple, but not simple. A semisimpleLie algebra L 6= 0 is not Abelian.

If L has a solvable Ideal I 6= 0 then L has an Abelian ideal 6= 0 too: Let i ∈Nbe the largest index from the derived series of I with DiI 6= 0. Then DiI ⊂ L is anAbelian ideal because Di+1I = [DiI,DiI] = 0. The reverse implication is obvious:Each Abelian ideal is solvable. As a consequence:

L semisimple ⇐⇒ rad(L) = 0.

The equivalence above indicates a complementarity between semisimplicity andsolvability. The only Lie algebra which is semisimple and also solvable is the zeroLie algebra L = 0.

A simple application of the above characterization is the following Proposition 4.6.

Proposition 4.6 (Semisimple Lie algebras are matrix algebras).

4.2 Fundamentals of semisimple Lie algebras 97

The adjoint representation of a semisimple Lie algebra is faithful. In particular,

L' ad(L)⊂ Der(L)

and L is isomorphic to a matrix algebra.

Proof. According to Proposition 2.8 the adjoint representations maps to the Liealgebra of derivations. The kernel is the center

ker[ad : L→ Der(L)] = Z(L)

which is an Abelian ideal of L. Therefore Z(L) = 0, and the adjoint representationis injective, q.e.d.

Proposition 4.7 (Semisimple Lie algebra sl(n,K)). The Lie algebra

sl(n,K) := A ∈ gl(n,K) : det A = 1

is semisimple.

Proof. i) Complex base field: Set L := sl(n,C) and R := rad(L). According toLie’s Theorem 3.21 we may assume that the solvable ideal R is an ideal of uppertriangular matrices

R⊂ t(n,C)∩L.

A direct conclusion from the definition of solvability is the equivalence

X ∈ R ⇐⇒ X> ∈ R.

Hence the elements from R are even diagonal:

R⊂ d(n,C)∩L.

Because R⊂ L is an ideal we have

[L,R]⊂ R.

For arbitrary X ∈ R we compute the commutator [E,X ] for elements E ∈ L from abasis of L: According to part i) the element X ∈ R is a diagonal matrix

X =n

∑j=1

x j j ·E j j.

For E = Ers with r 6= s

[X ,Ers] = ∑j

x j j · [E j j,Ers] = ∑j

x j j ·δ jr ·E js−∑j

x j j ·δ js ·Er j =


= xrr ·Ers− xssErs = (xrr− xss) ·Ers ∈ R⊂ d(n,C)

which implies xrr− xss = 0, hence

X = λ ·1,λ ∈ C,

is a scalar matrix. Because tr X = 0 we obtain λ = 0 and

X = 0.

ii) Real base field: A real Lie algebra M is semisimple iff its complexification

M⊗RC

is semisimple. According to this general result part i) implies the semisimplenessof sl(n,R), q.e.d.

Later we will see in Theorem 7.11 that the Lie algebras sl(n,C),so(n,C),sp(2n,C)are semisimple and - under the dimensional restrictions from Proposition 2.11 - evensimple Lie algebras. The most basic example from these series is the simple Lie al-gebra sl(2,C).

Any Lie algebra becomes semisimple after dividing out its radical.

Proposition 4.8 (Semisimpleness after dividing out the radical). For any Liealgebra L the quotient L/rad(L) is semisimple.

Proof. Consider the canonical morphism

π : L−→ L/rad(L).

Any idealI ⊂ L/rad(L)

has the formI = J/rad(L)

with the ideal J := π−1(I)⊂L. If I is an Abelian ideal, then [I, I] = 0, hence [J,J]⊂ rad(L).Therefore the derived algebra

D1J = [J,J]

is solvable. The latter property implies J solvable because

Di+1J = Di(D1J)⊂ Di(rad L) = 0.

As a consequence J ⊂ rad(L) and

I = π(J) = 0, q.e.d.


According to Proposition 4.8 any Lie algebra L fits into an exact sequence

0−→ I −→ L−→M −→ 0

with I = rad(L), a solvable subalgebra, and M = L/I semisimple. The Levi de-composition, Theorem ??, will show that one can even obtain L as the semi-directproduct

L' I oθ M.

Definition 4.9 (Orthogonal space of the trace form). Consider a K-Lie algebra Land a symmetric bilinear map

β : L×L→K.

i) The orthogonal space of a vector subspace M ⊂ L with respect to β is

M⊥ := x ∈ L : β (x,M) = 0.

The nullspace of β is L⊥.

ii) The form β is non-degenerate if it has trivial nullspace, i.e. L⊥ = 0.

Lemma 4.10 (Orthogonal space of ideals). Consider a vector space V , a Lie alge-bra L⊂ gl(V ) and the trace form

β : L×L→ C,β (x,y) := tr(xy).

For an ideal I ⊂ L the orthogonal space I⊥ ⊂ L of β is an ideal too.

Proof. Consider x ∈ I⊥. For arbitrary y ∈ L and u ∈ I

β ([x,y],u) = β (x, [y,u])

according to Lemma 4.1. Because [y,u] ∈ I and x ∈ I⊥

β (x, [y,u]) = 0.

Hence [x,y] ∈ I⊥, q.e.d.

The main step in characterizing semisimplicity of a Lie algebra by its Killingform is the following Lemma.

Lemma 4.11 (Non-degenerateness of the trace form for semisimple Lie alge-bras). Consider a vector space V and a semisimple Lie algebra L⊂ gl(V ). Thenthe trace form

β : L×L→ C,β (x,y) := tr(xy),

is non-degenerate.


Proof. The nullspace

S := L⊥ = x ∈ L : tr(xy) = 0 f or all y ∈ L

is an ideal according to Lemma 4.10. For x,y,z ∈ S

tr([x,y] z) = 0

because z ∈ S. Cartan’s trace criterion, see Theorem 4.4, applied to the Lie algebraS, shows that S is solvable. Hence

S⊂ L

is a solvable ideal. Semisimpleness of L implies S = 0, q.e.d.

The principal theorem of the present section is the following Cartan criterion forsemisimplicity. It derives from Cartan’s trace criterion for solvability.

Theorem 4.12 (Cartan criterion for semisimplicity). For a Lie algebra L the fol-lowing properties are equivalent:

• L is semisimple

• The Killing form κ of L is non-degenerate.

Proof. i) Assume L semisimple. According to Proposition 4.6 the adjoint represen-tations identifies L with the subalgebra

ad L⊂ gl(L).

Therefore κ is nondegenerate according to Lemma 4.11.

ii) Assume κ nondegenerate, i.e. with nullspace

S := L⊥ = 0.

Consider an Abelian idealI ⊂ L.

We claim I ⊂ S: For arbitrary, but fixed x ∈ I and all y ∈ L the composition

ad(x)ad(y) : L→ I ⊂ L

is a nilpotent endomorphism of L, because

Lad y−−→ L ad x−−→ I

ad y−−→ I ad x−−→ [I, I] = 0.

Hence


κ(x,y) = tr(ad(x)ad(y)) = 0.

As a consequence x ∈ S andI ⊂ S.

Now S = 0 implies I = 0. Therefore 0 is the only Abelian ideal of L, whichforces L to be semisimple, q.e.d.

Lemma 4.13 (Killing form of an ideal). Consider a Lie algebra L. For any ideal I ⊂ Lthe Killing form of I is the restriction of the Killing form of L to I

κI = κ|I× I.

Proof. Any base of the vector subspace I ⊂ L extends to a base of L. For x, y ∈ Ithe corresponding matrix representations(

A B0 0

)of (ad x) (ad y) : L→ I

and (A)

of the restriction (ad x)|I (ad y)|I : I→ I

showtr((ad x) (ad y)) = tr A = tr((ad x)|I (ad y)|I).

As a consequence

κ(x,y) = tr(ad(x)ad(y)) = tr((ad(x)ad(y))|I) = κI(x,y), q.e.d.

The first step on the way to split a semisimple Lie algebra as a direct sum ofsimple Lie algebras is the following Proposition 4.14.

Proposition 4.14 (Splitting a semisimple Lie algebra with respect to an ideal).Consider a semisimple Lie algebra L and an ideal I ⊂ L. Then:

1. The Lie algebra L decomposes as the direct sum of ideals

L = I⊕ I⊥.

2. Both ideals I ⊂ L and I⊥ ⊂ L are semisimple.

Note: Taking the direct sumI⊕ J

of two ideals of a Lie algebra presupposes that the sum of the underlying vectorspaces is direct. Then I∩ J = 0, which implies for the Lie bracket


[I,J]⊂ I∩ J = 0.

As a consequence: If a Lie algebra L, when considered as a vector space, splits asthe direct sum

L = I⊕ J

of the vector spaces underlying two ideals I, J ⊂ L, then L is isomorphic as Liealgebra to the direct product of the Lie algebras I and J:

L' I× J.

Proof. 1) Dimension formula: First we consider the underlying vector spaces. TheKilling form κ of L induces a map of vector spaces

F : L−→ L∗,x 7→ κ(x,−).

The map is injective because the Killing form is nondegenerate according to Car-tan’s criterion for semisimplicity, Theorem 4.12. It is also surjective because

dim L = dim L∗.

On one hand, for x ∈ I⊥ holds

F(x)(I) = 0, i.e. F(x)|I = 0.

On the other hand: Assume ϕ ∈ L∗ with ϕ|I = 0. Due to the surjectivity of F anelement x ∈ L exists with

ϕ = F(x) = κ(x,−).

Then ϕ|I = 0 implies x ∈ I⊥. Hence

F(I⊥) = ϕ ∈ L∗ : ϕ(I) = 0= (L/I)∗.

We obtain

dim I⊥ = dim F(I⊥) = dim (L/I)∗ = dim (L/I) = dim L−dim I,

i.e.dim L = dim I +dim I⊥.

Secondly, we show I ∩ I⊥ = 0: According to Lemma 4.10 also the orthogo-nal space I⊥ ⊂ L is an ideal. We apply Cartan’s trace criterion for solvability,Theorem 4.4, to the ideal

J := I∩ I⊥

considered as a Lie algebra. Denote by κ the Killing form of L and by κJ the Killingform of J. According to Lemma 4.13 the Killing form κJ is the restriction of κ . Wehave

κJ([J,J],J) = κ([J,J],J) = 0


because J⊂ I⊥ and [J,J]⊂ J⊂ I. The Cartan criterion implies that J⊂ L is solvable,and semisimpleness of L implies

J = 0.

As a consequence, the dimension formula for the sum of vector spaces

dim L = dim I +dim I⊥ = dim(I + I⊥)+dim (I∩ I⊥)

implies the direct sum decomposition

L = I⊕ I⊥.

2) Semisimpleness: Due to part 1) any Abelian ideal J ⊂ I is also an Abelian idealof L, hence

J = 0.

Analogously for an Abelian ideal of I⊥, q.e.d.

Theorem 4.15 (Splitting of a semisimple Lie algebra into simple summands).Consider a Lie algebra L.

1. The following properties are equivalent:

• L is semisimple

• L decomposes as a direct sum

L =⊕α∈A

Iα ,card A < ∞,

of simple ideals Iα ⊂ L.

2. Assume L semisimple.

i) Each ideal I ⊂ L is a finite direct sum of simple ideals. In particular, for eachideal I ⊂ L an ideal J ⊂ L exists with

L = I⊕ J.

ii) Each simple ideal I ⊂ L is one of the distinguished ideals Iα ,α ∈ A. In partic-ular, the representation of L as direct sum of simple ideals is uniquely determinedup to the order of the summands.

iii) The Lie algebra L equals its derived algebra, i.e.

L = D1L = [L,L].


Proof. 1. i) Assume L semisimple. In case L = 0 take the empty sum with A = /0.Otherwise L 6= 0 and L is not Abelian.

If L has no proper ideals, then L itself is simple. Otherwise we choose a properideal

0( I1 ( L

of minimal dimension. Proposition 4.14 implies a direct sum representation

L = I1⊕ I⊥1 .

The two ideals I1 and I⊥1 have the following properties:

• Any ideal of I1 or of I⊥1 is also an ideal of L because of the direct sum repre-sentation.

• The ideal I1 is simple, because any proper ideal of I1 would be a proper idealof L of dimension less than dim I1.

• The ideal I⊥1 is semisimple due to Proposition 4.14.

Continuing with I⊥1 the decomposition can be iterated until the last summand inthe direct sum representation

L = I1⊕ I2⊕ ...⊕ In

does not contain any proper ideal. The decomposition stops after finitely manysteps because each steps decreases the dimension of the ideals in question.

ii) Assume the direct sum decomposition of L

L =⊕α∈A

Iα ,card A < ∞.

For any Abelian ideal I ⊂ L and α ∈ A the intersection

I∩ Iα

is an Abelian ideal of the simple Lie algebra Iα . Hence either

I∩ Iα = 0

orI∩ Iα = Iα .

The latter case is excluded because the simple Lie algebra Iα is not Abelian. Asa consequence I = 0.

2. If L is semisimple then we may presuppose the direct sum decomposition of Laccording to part 1).

i) and ii) Ideal: The identity


idL : L−→⊕α∈A

Iα

induces for any α ∈ A the projection morphism

pα : L−→ Iα ,

andidL =

⊕α∈A

pα .

If I 6= 0 then0 6= I = p(I) =

⊕α∈A

pα(I).

Hence for at least one α ∈ A

pα(I) 6= 0.

Now pα(I)⊂ Iα is an ideal because

[Iα , pα(I)] = [pα(L), pα(I)] = pα([L, I])⊂ pα(I).

Simpleness of Iα impliespα(I) = Iα .

Thenpα([Iα , I]) = [pα(Iα), p(I)] = [Iα , Iα ] 6= 0

because the simple ideal Iα is not Abelian. Hence

0 6= [Iα , I]⊂ Iα ,

which implies[Iα , I] = Iα

and eventuallyIα = [Iα , I]⊂ I.

If I is simple, thenIα = I.

For a general ideal I ⊂ L we just proved

pα(I) 6= 0 =⇒ Iα ⊂ I.

For any x ∈ I we have

x = ∑α∈A

pα(x) ∈⊕

pα (x)6=0

Iα .


HenceI =

⊕α∈A′

Iα , A′ = α ∈ A : pα(I) 6= 0.

iii) Derived algebra: The splitting of L implies the direct sum decomposition

[L,L] = [⊕α∈A

Iα ,⊕β∈A

Iβ ] =⊕

α,β∈A

[Iα , Iβ ] =⊕α∈A

[Iα , Iα ].

For each α ∈ A either [Iα , Iα ] = 0 or [Iα , Iα ] = Iα because the simple Lie alge-bra Iα has no proper ideals. The case [Iα , Iα ] = 0 is excluded because Iα is notAbelian. As a consequence

[L,L] =⊕α∈A

Iα = L, q.e.d.

4.3 Weyl’s theorem on complete reducibility of representations

Alike to splitting a semisimple Lie algebra into a direct sum of simple Lie alge-bras Weyl’s Theorem 4.24 splits an arbitrary finite-dimensional representation of asemisimple Lie algebra L as a direct sum of irreducible representations of L.

Definition 4.16 (Reducible and irreducible modules). Consider a K-Lie algebra L,a vector space V , and a representation of L

ρ : L→ gl(V ).

According to Definition 2.4 the latter is named an L-module with respect to ρ . Asa shorthand one often uses for the module operation the notation from commutativealgebra

L×V →V,(x,v) 7→ v := x ·v := ρ(x)(v).

1. A submodule W of V is a subspace W ⊂V stable under the action of L, i.e.

ρ(L)(W )⊂W.

2. An L-module V is irreducible iff V has exactly the two L-submodules V and 0.Otherwise V is reducible. Notably, the zero-module is reducible.

3. A submodule W of V has a complement iff a submodule W ′ ⊂V exists with

V =W ⊕W ′

as a direct sum of vector spaces.

4.3 Weyl’s theorem on complete reducibility of representations 107

4. An L-module V is completely reducible iff a decomposition exists

V =k⊕

j=1

Wj

with irreducible L-modules Wj, j = 1, ...,k.

Applying the standard constructions from linear algebra to L-modules generatesa series of new L-modules based on existing L-modules.

Definition 4.17 (Induced representations). Consider a Lie algebra L. Two repre-sentations of L

ρ : L→ gl(V ) and σ : L→ gl(W )

with corresponding L-modules V and W induce further representations of L in acanonical manner:

1. Dual representation ρ∗ : L→ gl(V ∗) with - note the minus sign -

(ρ∗(x)λ )(v) :=−λ (ρ(x)v), (x ∈ L,λ ∈V ∗,v ∈V ).

Corresponding module: Dual module V ∗ with

(x.λ )(v) =−λ (x.v).

2. Tensor product ρ⊗σ : L→ gl(V ⊗W ) with

(ρ⊗σ)(x)(v⊗w) := ρ(x)v⊗w+ v⊗σ(x)w, (x ∈ L,v ∈V,w ∈W ).

Corresponding module: Tensor product V ⊗W with

x.(u⊗v) = x.u⊗v+u⊗ x.v.

3. Exterior product ρ ∧ρ : L→ gl(∧2 V ) with

(ρ ∧ρ)(x)(v1⊗v2) := ρ(x)v1∧v2 +v1∧ρ(x)v2, (x ∈ L,v1,v2 ∈V ).

Corresponding module: Exterior product∧2 V with

x.(u∧v) = x.u∧v+u∧ x.v.

4. Symmetric product Sym2(ρ) : L→ gl(Sym2V ) with

(Sym2(ρ)(x))(v1 ·v2) := (ρ(x)v1) ·v2 +v1 · (ρ(x)v2), (x ∈ L,v1,v2 ∈V ).

Corresponding module: Symmetric product Sym2V with

x.(u ·v) = (x.u) ·v+u · (x.v).


5. Hom-representation HomK(ρ,σ) := τ : L→ gl(HomK(V,W )) with

(τ(x) f )(v) := σ(x) f (v)− f (ρ(x)v), (x ∈ L, f ∈ HomK(V,W ),v ∈V ).

Corresponding module: Module of homomorphisms HomK(V,W ) with

(x. f )(v) = x. f (v)− f (x.v).

The constructions from part 2) - 4) generalize to products of more than two fac-tors. The notations emphasize the tight relationship to similar constructions in Com-mutative Algebra for modules over rings.

Remark 4.18 (Induced representations).

1. It remains to check that the constructions in Definition 4.17 actually yield L-modules. As an example we consider the case of the dual module V ∗ and verifythat the induced map

ρ∗ : L−→V ∗

satisfies(ρ∗(x)λ )(v) :=−λ (ρ(x)v)

or(x.λ )(v) =−λ (x.v)

preserves the Lie bracket. Claim: For all x,y ∈ L,λ ∈V ∗ and v ∈V

([x,y].λ )(v) = (x.y.λ − y.x.λ )(v)

Left-hand side:

([x,y].λ )(v) =−λ ([x,y].v) =−λ (x.y.v)+λ (y.x.v).

Right-hand side:

(x.y.λ − y.x.λ )(v) =−y.λ (x.v)+ x.λ (y.v) = λ (y.x.v)−λ (x.y.v).

2. One can show that the canonical isomorphism of vector spaces

V ∗⊗W '−→ HomK(V,W ),λ (−)⊗w 7→ λ (−) ·w,

is an isomorphism of L-modules.

3. For a K-linear map f ∈ HomK(V,W ):

L. f = 0 ⇐⇒ f : V →W is L-linear.


Theorem 4.19 (Lemma of Schur). Consider a K-Lie algebra L.

1. Then any morphismf : V −→W

between two irreducible L-modules is either zero or an isomorphism.

2. If K= C andρ : L→ gl(V )

is an irreducible complex representation, then any endomorphism f ∈ End(V )commuting with all elements of ρ is a multiple of the identity, i.e. if for all x ∈ L

[ f ,ρ(x)] = 0

then a complex number µ ∈ C exists with

f = µ · id.

3. If K= C, andf1, f2 : V →W

are two morphisms of irreducible L-modules with f2 6= 0. Then

f1 = c · f2

for a suitable c ∈ C, i.e.

HomL(V,W ) =

C V 'W0 V 6'W

Proof. 1. Because f is a morphism, the kernel ker f ⊂ V is a submodule. Irre-ducibility of V implies that ker f = 0 or ker f = V . If ker f = 0 then f isinjective and f (V )⊂W is a submodule. Irreducibility of W implies f (V ) = W ,such that f is injective and surjective, hence an isomorphism.

2. The endomorphism f has a complex eigenvalue µ . Its eigenspace W ⊂V is alsoan L-submodule: If w ∈W and

f (w) = c ·w

then for all x ∈ L

( f ρ(x))(w) = (ρ(x) f )(w) = ρ(x)(c ·w) = c ·ρ(x)w.

Now W 6= 0 and V irreducible imply W =V .

3. Because of f2 6= 0 the morphism f2 is an isomorphism according to part 1). Con-sider the L-module morphism


f := f1 f−12 : W −→W.

Then f ∈ End(W ), and for all x ∈ L, w ∈W, holds

x. f (w) = f (x.w),

which implies due to part 2) the existence of a suitable c ∈ C with

f = c · idW

which provesf1 = c · f2.

The last claim about HomL(V,W ) follows from part 1), q.e.d.

Definition 4.20 (Casimir element). Consider a complex semisimple Lie algebra Land a faithful representation ρ : L→ gl(V ) on a complex vector space V . The traceform of ρ

β : L×L→ C,β (x,y) := tr(ρ(x)ρ(y)),

is nondegenerate according to Lemma 4.11. For a base (xi)i=1,...,n of L denoteby (y j) j=1,...,n the base dual with respect to β , i.e.

β (xi,y j) = δi j.

The Casimir element of ρ is defined as the endomorphism

cρ :=n

∑i=1

ρ(xi)ρ(yi) ∈ End(V ).

Note: One checks that the Casimir element does not depend on the choice of thebasis (xi)i=1,...,n.

Remark 4.21 (Reduction to faithful representations). If the representation ρ is notfaithful one considers the direct decomposition

L = ker ρ⊕L′

with L′ :=(ker ρ)⊥⊂L. The Lie algebra L′ is semisimple according to Proposition 4.14.The restricted representation ρ|L′→ gl(V ) is faithful.


Theorem 4.22 (Properties of the Casimir element). The Casimir element cρ ∈ End(V )of a faithful representation ρ of a complex semisimple Lie algebra L has the follow-ing properties:

• Commutation: The Casimir element commutes with all elements of the represen-tation:

[cρ ,ρ(L)] = 0

• Trace: tr(cρ) = dim L

• Scalar: For an irreducible representation ρ holds

cρ =dim Ldim V

· idV .

Proof. Set n = dim L and

cρ =n

∑i=1

ρ(xi)ρ(yi) ∈ End(V )

with bases (xi)i=1,...,n and (y j) j=1,...,n of L, which are dual with respect to the traceform β of ρ .

i) Commutation: For x ∈ L we show [ρ(x),cρ ] = 0: Define the coefficients (ai j)and (b jk) according to

[x,xi] =n

∑j=1

ai j · x j, [x,y j] =n

∑k=1

b jk · yk.

Because the families (x j)1≤ j≤n and (yk)1≤k≤n are dual bases with respect to β

ai j = β (n

∑k=1

aik · xk,y j) = β ([x,xi],y j) =−β ([xi,x],y j) =−β (xi, [x,y j]) =

=−β (xi,n

∑k=1

b jk · yk) =−b ji.

Here we made use of the associativity of the trace form according to Lemma 4.1. Tocompute

[ρ(x),cρ ] =n

∑i=1

[ρ(x),ρ(xi)ρ(yi)]

we use the formula[A,BC] = [A,B]C+B[A,C]

for endomorphisms A,B,C ∈ End(V ). For each summand

[ρ(x),ρ(xi)ρ(yi)] = [ρ(x),ρ(xi)]ρ(yi)+ρ(xi)[ρ(x),ρ(yi)]


and therefore

[ρ(x),cρ ] =n

∑i=1

([ρ(x),ρ(xi)]ρ(yi)+ρ(xi)[ρ(x),ρ(yi)]) =

=n

∑i=1

(ρ([x,xi])ρ(yi)+ρ(xi)ρ([x,yi]) =n

∑i, j=1

ai j ·ρ(x j)ρ(yi)+n

∑i,k=1

bik ·ρ(xi)ρ(yk) =

=n

∑i, j=1

ai j ·ρ(x j)ρ(yi)+n

∑i, j=1

b ji ·ρ(x j)ρ(yi) =

=n

∑i, j=1

(ai j +b ji) · (ρ(x j)ρ(yi)) = 0.

Here we have changed in the second sum the summation indices (i,k) 7→ ( j, i).

ii) Trace: We have

tr(cρ) =n

∑i=1

tr(ρ(xi)ρ(yi)) =n

∑i=1

β (xi,yi) = n = dim L.

iii) Scalar: For an irreducible representation ρ we get with part i) and the Lemmaof Schur, Theorem 4.19,

cρ = µ · idV

and with part ii)tr(cρ) = µ ·dim V = dim L, q.e.d.

Lemma 4.23 (Representations of semisimple Lie algebras are traceless). Con-sider a complex semisimple Lie algebra L and a representation ρ : L→ gl(V ).

• Then ρ(L)⊂ sl(V ), i.e. tr(ρ(x)) = 0 for all x ∈ L.

• In particular, 1-dimensional representations of L are trivial, i.e. if dim V = 1then ρ = 0.

Proof. Due to semisimpleness we have L= [L,L], see Theorem 4.15. If x= [u,v]∈ Lthen

tr(ρ(x)) = tr(ρ([u,v])) = tr([ρ(u),ρ(v)]) = 0

according to Lemma 4.1. For 1-dimensional V , i.e. V = C, and for all x ∈ L

0 = tr(ρ(x)) = ρ(x), q.e.d.


Theorem 4.24 (Weyl’s theorem on complete reducibility). Every finite-dimensional,non-zero L-module V of a complex semisimple Lie algebra L is completely reducible.

Note: The irreducible direct summands of V are uniquely determined, not only theirisomorphy classes.

Proof. We have to show: Any submodule of a L-module has a complement. Hencewe will consider pairs (V,W ) with an L-module V and a submodule W ⊂V proceed-ing along the following steps:

1. codimVW = 1. Proof by induction on n = dim W .

- Subcase 1a): W reducible.

- Subcase 1b): W irreducible.

2. codimVW arbitrary.

1. Codimension = 1: Consider pairs (V,W ) with an L-module V and a submodule

W ⊂V with codimVW = 1.

Lemma 4.23 implies that the 1-dimensional quotient V/W is a trivial L-module.Hence (V,W ) fits into an exact sequence of L-modules

0→W →V → C→ 0

We construct a complement of W by induction on dim W . The induction step em-ploys one of two alternative subcases. Subcase 1a) uses the induction hypothesis,while subcase 1b) uses the Casimir element.

Subcase 1a), W reducible: Consider a proper submodule

0(W ′ (W,

inducing the exact sequence of L-modules

0→W/W ′→V/W ′→ C→ 0

Becausedim (W/W ′)< dim W

the pair (V/W ′,W/W ′) satisfies the induction assumption. We obtain a submod-ule W ′ ⊂ W ⊂V and a splitting of L-modules

V/W ′ =W/W ′⊕W/W ′.

We have the exact sequence of L-modules


0→W ′→ W → W/W ′ ' C→ 0

The estimatescodimWW ′ = 1 and codimW ′W ≥ 1

implydim W ′ < dim W

also the pair (W ,W ′) satisfies the induction assumption. We obtain a submodule X ⊂ Wand a splitting of L-modules

W =W ′⊕X .

We claimV =W ⊕X .

For the proof, on one hand the two splittings imply

dim V = dim W +dim W −dim W ′ and dim W = dim W ′+dim X .

Hencedim V = dim W +dim X .

On the other hand,X ⊂ W

(W ∩X)⊂ (W ∩W ).

The first splitting implies

0= (W/W ′)∩ (W/W ′) = (W ∩W )⊂W ′,

and therefore(W ∩X)⊂ (W ∩W )⊂W ′.

As a consequence

W ∩X = (W ∩X)∩W ′ =W ∩ (X ∩W ′).

The second splitting implies

X ∩W ′ = 0.

HenceW ∩X = 0.

ThereforeV =W ⊕X

which finishes the induction step for reducible W .

Subcase 1b), W irreducible: Assume that the representation


ρ : L→ gl(V )

defines the L-module structure. According to Remark 4.21 we may assume ρ

faithful. We consider the Casimir element of ρ

cρ :=dim L

∑j=1

ρ(x j)ρ(y j) ∈ End(V ).

• Concerning V : The Casimir element cρ :V→V is even L-linear, i.e. for x ∈V, v ∈V ,

(cρ ρ(x))(v) = (ρ(x) cρ)(v)

because [cρ ,ρ(x)] = 0 due to Theorem 4.22. Therefore

X := ker cρ ⊂V

is an L-submodule.

• Concerning W : Because the 1-dimensional L-module V/W is trivial, we have

ρ(L)(V )⊂W.

As a consequence, the definition of cρ implies that the submodule W ⊂ V isstable also with respect to cρ , i.e.

cρ(W )⊂W.

The irreducibility of W implies according to Theorem 4.22:

cρ |W = µ · idW .

• Concerning W ⊂V : If we extend a basis (w1, ...,wn) of the vector subspace Wto a basis B = (w1, ...,wn,v) of the vector space V , then with respect to B theCasimir element has the matrix

cρ =

µ ∗

. . .µ ∗

0 ... 0 0

From

tr cρ = dim L

follows µ 6= 0, and in particular

W ∩X = 0.

From det cρ = 0 follows


X = ker cρ 6= 0.

Because codimVW = 1 we get

V =W ⊕X

which finishes the induction step for irreducible W .

2. Arbitrary codimension: Consider an arbitrary proper submodule

0(W (V.

First we claim the existence of a section against the canonical injection

W −→V,

i.e. the existence of an L-linear map

f0 : V →W

such that f0|W = idW . ThenX := ker f0

is a complement of W because V/X 'W .

The idea is to translate the question on the existence of a section to a problemabout L-modules in a context where case 1 applies. We consider the induced L-module

HomC(V,W )

to reduce the question on sections to a problem concerning pairs of L-modulesof C-linear homomorphism

(V ,W )

with codimV W = 1. The latter problem can be solved by case 1.

Consider the following submodules of the L-module HomC(V,W )

V := f ∈ HomC(V,W ) : f |W = λ · idW ,λ ∈ C

W := f ∈ V : f |W = 0.

In order to prove that V is a L-module, consider f ∈HomC(V,W ) with f |W = λ · idWand x ∈ L,w ∈W :

(x. f )(w) = x. f (w)− f (x.w) = x. f (λ ·w)−λ · (x.w) = λ · (x.w)−λ · (x.w) = 0.

Therefore even L.V ⊂W and both V and W are L-modules. By definition

codimV W = 1


because W as submodule of V is defined by one additional linear equation.

We now apply the result of case 1 to the pair (V ,W ):

The hyperplane W ⊂ V has a complement

C · f0, f0 ∈ V ⊂ HomC(V,W ),

i.e.V = W ⊕C · f0.

Without restrictionf0|W = idW .

The L-module C · f0 is 1-dimensional, hence trivial according to Lemma 4.23.According to Remark 4.18 the equality L. f0 = 0 implies the L-linearity of

f0 : V →W.

ThereforeX := ker f0 ⊂V

is a submodule. Due to our considerations at the beginning of case 2)

V =W ⊕X , q.e.d.

Note. Theorem 4.24 also holds over the base field R.

The following Theorem 4.25 is a consequence of Theorem 4.24.

Theorem 4.25 (Jordan decomposition in semisimple matrix Lie algebras). Con-sider a vector space V and a semisimple complex matrix Lie algebra L⊂ gl(V ). Forany element x ∈ L, considered as endomorphism

x : V →V

with Jordan decomposition

x = xs + xn ∈ End(V ),

both summands belong to L, i.e. xs,xn ∈ L.

Proof. The proof has two separate parts. The first part deals with the Lie algebra Land considers L ⊂ gl(V ) as a matrix algebra. The second part deals with the Liealgebra L when considered an abstract semisimple Lie algebra.

We consider an arbitrary, but fixed endomorphism


x : V →V

with Jordan decompositionx = xs + xn.

i) The Lie algebra gl(V ): Here adjoint elements are taken with respect to gl(V ).Proposition 3.3 implies the Jordan decomposition

ad x = ad xs +ad xn

with polynomials ps(T ), pn(T ) ∈ C[T ] without constant term such that

ad xs = ps(ad x),ad xn = pn(ad x).

Therefore (ad x)(L)⊂ L implies

[xs,L] = (ad xs)(L)⊂ L and [xn,L] = (ad xn)(L)⊂ L.

The normalizer of the subalgebra L⊂ gl(V )

N := Ngl(V )L⊂ gl(V ).

contains L by definition as an ideal. We just proved

xs,xn ∈ N.

ii) The semisimple Lie algebra L: We have xs,xn ∈N. But unfortunately, in general L ( N.Therefore we will construct a specific submodule

L⊂ L⊂ N

with the property xs,xn ∈ L and show L = L.

For any submodule W ⊂V of the L-module V we define the subspace

LW := y ∈ gl(V ) : y(W )⊂W and tr(y|W ) = 0 ⊂ HomC(V,V ).

One checks that LW is even an L-submodule of the induced L-module HomC(V,V ):For z ∈ L, y ∈ LW , w ∈W holds

(z.y)(w) = z.y(w)− y(z.w) ∈W

andtr((z.y)|W ) = tr([z,y]|W ) = 0.

We defineL := N∩

⋂W⊂V

LW =⋂

W⊂V

(N∩LW ).

With N and each LW also L is an L-module. It satisfies:


• L ⊂ L: Because L is semisimple, Lemma 4.23 implies tr(y|W ) = 0 for all y ∈ L.As a consequence

L⊂⋂

W⊂V

LW and L⊂ L.

• xs,xn ∈ L: Because the subspace W is stable with respect to the endomorphism x : V →V ,the same is true for its Jordan components, i.e.

xs(W )⊂W and xn(W )⊂W.

According to Lemma 4.23, the semisimpleness of L implies tr(x|W )= 0. Hence x ∈ LW .With xn also the restriction xn|W is nilpotent, therefore

tr (xn|W ) = 0 and xn ∈ LW .

As a consequence alsoxs = x− xn ∈ LW .

We obtainxs,xn ∈ L

because xs,xn ∈ N and xs,xn ∈ LW for all L-submodules W ⊂V .

It remains to show L = L. According to Weyl’s theorem on complete reducibility,Theorem 4.24, an L-submodule M ⊂ L exists with

L = L⊕M.

We claim: M = 0. Consider an arbitrary but fixed y ∈M. Because L⊂ N we have

[L,L] = [L, L]⊂ L

which implies that the action of L on M is trivial. In order to show M = 0 weconsider an arbitrary, but fixed y ∈M. Then

[L,y] = 0.

As a consequence, for any irreducible L-submodule W ⊂V on one hand, Schur’sLemma, Theorem 4.19, implies

y = µ · idW .

On the other hand, y ∈ LW implies tr(y|W ) = 0, hence

y|W = 0.

The decomposition of V as a direct sum of irreducible L-modules shows y = 0. Weobtain M = 0 and


L = L, q.e.d.

The previous Theorem 4.25 assures: For a semisimple Lie algebra L the Jordandecomposition of elements from ad L induces a decomposition of elements from L.

Definition 4.26 (Abstract Jordan decomposition). Consider a complex semisim-ple Lie algebra L. Its adjoint representation satisfies

ad : L '−→ ad L⊂ gl(L).

For any x ∈ L the endomorphism

ad x : L→ L,y 7→ [x,y]

has the Jordan decomposition

ad x = fs + fn ∈ End(L)

withfs ∈ ad(L) and fn ∈ ad(L),

see Theorem 4.25. Defining

s := ad−1( fs) ∈ L and n := ad−1( fn) ∈ L

provides the decompositionx = s+n

with s ∈ L ad-semisimple, n ∈ L ad-nilpotent and [s,n] = 0. This decomposition isnamed the abstract Jordan decomposition of x ∈ L.

Note. The abstract Jordan decomposition

x = s+n

is uniquely determined by the property, that the components s, n∈ L are respectivelyad-semisimple and ad-nilpotent and satisfy [s,n] = 0. The uniqueness follows fromthe uniqueness of the Jordan decomposition of the endomorphism

ad x : L−→ L

and from the isomorphismad : L '−→ ad(L).


In the abstract case, neither x ∈ L nor its components s, n ∈ L from the abstractJordan decomposition are defined as endomorphisms of a vector space. But Corol-lary 4.27 will show: For any representation

ρ : L→ gl(V )

the abstract Jordan decomposition of x ∈ L induces the Jordan decompositionof ρ(x) ∈ End(V ). This fact generalizes Proposition 3.3.

Corollary 4.27 (Jordan decomposition of representations). Consider a complexvector space V and a representation

ρ : L→ gl(V )

of a complex semisimple Lie algebra L. If x ∈ L has the abstract Jordan decomposi-tion

x = s+n

then ρ(x) ∈ End(V ) has the Jordan decomposition

ρ(x) = ρ(s)+ρ(n).

Proof. i) Abstract Jordan decomposition in ρ(L): We consider the semisimple com-plex matrix Lie algebra

F := ρ(L)⊂ gl(V )

and the elementρ(x) ∈ F.

By definition the endomorphism

ad s : L→ L

is semisimple, i.e. the eigenvectors (vi)i∈I of ad s∈ End(L) span the vector space L.Hence the vectors from ρ(L)

(ρ(vi))i∈I

span the vector space F . They are eigenvectors of ad(ρ(s)) because

(ad s)(vi) = [s,vi] = λi ·vi

implies

(ad(ρ(s))(ρ(vi)) = [ρ(s),ρ(vi)] = ρ([s,vi]) = ρ(λi ·vi) = λi ·ρ(vi).

Therefore the endomorphism

ad ρ(s) ∈ End(F)


is semisimple. Similarly, the endomorphism

ad n ∈ End(L)

is nilpotent by definition, and therefore

ad ρ(n) ∈ End(F)

is nilpotent because(ad ρ(n))k = ρ (ad n)k.

Apparently,[ρ(s),ρ(n)] = ρ([s,n]) = 0

which implies [ad ρ(s),ad ρ(n)] = 0. Summing up:

ad ρ(x) = ad ρ(s)+ad ρ(n)

is the Jordan decomposition of the element

ad ρ(x) ∈ ad(F).

By definitionρ(x) = ρ(s)+ρ(n) ∈ F

is the abstract Jordan decomposition in the Lie algebra F of the element ρ(x) ∈ F .

ii) Jordan decomposition in End(V ): Now we consider

ρ(x) : V −→V

as endomorphism of the vector space V . We denote its Jordan decomposition due toTheorem 1.15 by

ρ(x) = fs + fn.

Because the matrix Lie algebra F ⊂ gl(V ) is semisimple, Theorem 4.25 applies:

fs, fn ∈ ρ(L).

Semisimpleness of fs implies ad-semisimpleness, see proof of Proposition 3.3. Andnilpotency of fn implies ad-nilpotency, see Lemma 3.2. Hence also

ρ(x) = fs + fn

is the abstract Jordan decomposition of ρ(x) ∈ F . From the uniqueness of the ab-stract Jordan decomposition in F derives

fs = ρ(s) and fn = ρ(n), q.e.d.

Part IIComplex Semisimple Lie Algebras

124

Which are the fundamental results from the theory of complex semisimpleLiealgebras?

Structure of complex semisimple Lie algebras L

1. Cornerstone sl(2,C): The structure of the semisimple Lie algebra sl(2,C) due tothe basis (h,x,y) with commutators

[h,x] = 2x, [h,y] =−2y, [x,y] = h

generalizes to the Cartan decomposition of L, see Definition 5.17,

L = H⊕⊕

α∈Φ+

(Lα ⊕L−α)

with 1-dimensional root spaces

Lα = C · xα , L−α = C · yα

satisfying[xα ,yα ] = hα ∈ H.

For each root α ∈Φ+ the Lie algebra

Sα := span < hα , xα , yα >

is isomorphic to sl(2,C), see Proposition 7.3.

2. Integrality of the roots: For the first time the importance of integer numbers forcomplex Lie algebras shows up as the integer = 2 in the commutator relations

[h,x] = 2x and [h,y] =−2y.

Via the adjoint representation the element h ∈ sl(2,C) acts as diagonal endomor-phism on sl(2,C), and the corresponding eigenvalues, are

0,2,−2.

In the general case, under the adjoint representation the Cartan algebra H ⊂ Lacts in a diagonal way on L, and the corresponding eigenvalues, the roots fromthe root set Φ = Φ+∪Φ−,

α : H −→ C

satisfy< α1,α2 >:= α1(hα2) ∈ Z (Cartan integer),

see Proposition 7.4. The proof relies on the existence of the subalgebras

Sα ' sl(2,C)⊂ L

125

and the integer weights of finite-dimensional sl(2,C)-modules.

3. Root system: The integrality of the Cartan integers proves that the root set Φ

of L is a root system, see Definition 6.2: First, the non-degenerateness of theKilling-form κ provides an identification of the Cartan algebra H ⊂ L with itsdual space

H '−→ H∗, tα 7→ α,

and introduces on H∗ the non-degenerate bilinear form

(α,β ) := κ(tα , tβ ),

see Definition 7.1.

Secondly, the root set Φ spans the dual space H∗. After choosing a basis of H∗

formed by roots, any other root is a rational linear combination of the basis roots.The restriction of the form (−,−) to the rational vector space

VQ := spanQΦ

and to its real extensionV :=VQ⊗QR

is a scalar product, see Proposition 7.4. As a consequence the root set Φ ⊂ Vsatisfies all properties of a root system, see Definition 6.2, with Weyl reflections

σα : V −→V,v 7→ v−< v,α > ·α, < v,α >:= 2 ·(v,α)

(α,α)= v(hα),

which leave the scalar product invariant.

A highlight of the theory is the classification of root systems by their Dynkindiagrams, see Proposition 6.27.

4. Classification: From the classification of Dynkin diagrams results the classifica-tion of complex simple Lie algebras. There are finitely many types: The A,B,C,D-seriesof the classical Lie algebras, see Section 7.2, and in addition finitely many ex-ceptional types E,F,G. Any semisimple complex Lie algebra L is a finite directsum of simple Lie algebras, see Theorem 4.15.

Finite-dimensional L-modules

1. Complete reducibility: Thanks to Weyl’s theorem on complete reducibility eachfinite dimensional representation splits into a finite direct sum of irreduciblerepresentations, see Theorem 4.24.

2. Universal enveloping algebra: One obtains an irreducible sl(2,C)-module V ofarbitrary highest weight λ ∈ N by taking an (1+λ )-dimensional vector space Vwith basis (ei)i=0,...,λ and defining

• h.ei := (λ −2i) · ei• y.ei := (i+1) · ei+1, eλ+1 := 0• x.ei := (λ − i+1) · ei−1, e−1 := 0,

see Theorem 5.8. Here e0 ∈V is a primitive element.

To generalize this result to L-modules one introduces for any Lie algebra A theuniversal enveloping algebra UA of A as a quotient of the tensor algebra of A,see Definition 8.1. To obtain an irreducible UL-module V of arbitrary highestweight λ ∈ H∗ one takes an arbitrary non-zero vector e and sets

h.e = λ (h) · e for all h ∈ H,

andx.e = 0 for all α ∈Φ

+ and x ∈ Lα .

The 1-dimensional representation Ce of the Borel subalgebra

B := H⊕⊕

α∈Φ+

Lα ⊂ L

can be considered an UB-module. It extends to the UL-module

UL⊗UBCe.

Its quotient V by the sum of its proper submodules is irreducible, as UL-moduleand also as L-module, see Theorem 8.21.

3. Monoid of finite-dimensional irreducible L-modules: An irreducible L-module Vwith highest weight λ ∈ H∗ is finite-dimensional if and only if λ is integral andnon-negative.To show that λ is integral and non-negative one uses the subalgebras Sα ' sl(2,C).The opposite direction follows from the transitive action of the Weyl group of Lon the weights of V , see Theorem 8.23. As a consequence the Abelian monoid offinite-dimensional irreducible L-modules and the zero-module is free with basisthe fundamental weights

(λi)i=1,...,r, r = rank L,

see Theorem 8.26. The fundamental weights are obtained from a base

∆ = α1, ...,αr

of the root system of L by applying the Cartan matrix, see Proposition 8.25.

Chapter 5Cartan decomposition

The first non-trivial Cartan decomposition of a complex semisimple Lie algebra isthe decomposition of sl(2,C) in Proposition 5.4.

sl(2,C) = C ·h⊕ (C · x⊕C · y)

with h ∈ sl(2,C) ad-semisimple and the elements x,y ∈ sl(2,C) eigenvectors of hunder the adjoint representation.

The base field in this chapter is K = C, the field of complex numbers. All Liealgebras are complex Lie algebras unless stated otherwise.

5.1 Toral subalgebra

Consider a Lie algebra L. We recall from Definition 3.4 that an element x ∈ L isnamed ad-semisimple iff the endomorphism

ad x : L−→ L, y 7→ [x,y],

is semisimple, i.e. diagonalizable.

It is well-known from Linear Algebra that a set of commuting semisimple en-domorphisms can be simultaneously diagonalized. Therefore, starting from one ad-semisimple of L one tries to find all pairwise commuting ad-semisimple elementsof L which also commute with the given one. A first result about a semisimple Liealgebra L states: Any subalgebra T ⊂ L with only ad-semisimple elements is alreadyAbelian, see Proposition 5.3. Such a subalgebra is called a toral subalgebra.

A second result, Theorem 5.16 from Section 5.3, will show that no other elementsof L commute with a maximal toral subalgebra T ⊂ L, i.e. CL(T ) = T .

127

128 5 Cartan decomposition

Definition 5.1 (Toral subalgebra). Consider a Lie algebra L.

• A toral subalgebra of L is a subalgebra T ⊂ L with all elements x ∈ T ad-semisimple.

• A toral subalgebra T ⊂ L is a maximal toral subalgebra iff T is not properlycontained in any other toral subalgebra of L.

We recall from Linear Algebra the following Lemma 5.2: The restriction of adiagonalizable endomorphism to an invariant subspace stays diagonalizable.

Lemma 5.2 (Restriction of a diagonalizable endomorphism to a stable sub-space). Consider a vector space V and a diagonalizable endomorphism

x : V →V

inducing the eigenspace decomposition

V =⊕

λ

V λ (x).

If W ⊂V is a stable subspace, i.e. x(W )⊂W, then also the restriction

x|W : W →W

is diagonalizable with eigenspace decomposition

W =⊕

λ

(W ∩V λ (x)).

Proof. By assumption V decomposes as a direct sum of eigenspaces of x

V =k⊕

i=1

V λi(x)

with pairwise distinct eigenvalues λi, i = 1, ...,k. In order to prove

W =W ∩k⊕

i=1

V λi(x) =k⊕

i=1

(W ∩V λi(x))

we show: Ifw = v1 + ...+vk ∈W

with eigenvectors vi of x with pairwise distinct eigenvalues λi, then vi ∈ W forall i = 1, ...,k.

The claim holds for k = 1. For the induction step k−1 7→ k consider

w = v1 + ...+vk ∈W

with

5.1 Toral subalgebra 129

x(w) = λ1 ·v1 + ...+λk ·vk ∈W

and apply the induction assumption to

x(w)−λ1 ·w = (λ2−λ1) ·v2 + ...+(λk−λ1) ·vk ∈W.

We obtain v2, ...,vk ∈W . The representation

w = v1 + ...+vk ∈W

implies v1 ∈W , q.e.d.

We prove that the abstract Jordan decomposition of a semisimple Lie algebra L,cf. Definition 4.26, implies the existence of a maximal toral subalgebra of L.

Proposition 5.3 (Toral subalgebras). Consider a semisimple Lie algebra L 6= 0.

1. The Lie algebra L has a non-zero toral subalgebra, hence also a nonzero maximaltoral subalgebra.

2. Each toral subalgebra of L is Abelian.

Proof. 1. Existence of non-zero toral subalgebras: Not every element x ∈ L is ad-nilpotent. Otherwise ad L and also L' ad L were nilpotent according to Engel’stheorem, see Theorem 3.10. But the property [L,L] = L, see Theorem 4.15, showsthat L is not nilpotent.Choose an element x ∈ L with ad xs 6= 0. The abstract Jordan decomposition

x = s+n

provides a non-zero ad-semisimple element s∈ L. The 1-dimensional subalgebra

C · s⊂ L

is a toral subalgebra.

2. Toral subalgebras are Abelian: Consider a toral subalgebra T ⊂ L. Choosing anarbitrary but fixed x ∈ T we have to show: The restriction

adT (x) := ad(x)|T : T → T,y 7→ ad(x)(y) = [x,y],

is the zero morphism.

Assume an arbitrary element y ∈ T, y 6= 0. Due to Lemma 5.2 semisimplenessof ad(x) implies semisimpleness of adT (x). Hence we may assume y ∈ T as aneigenvector of adT (x) belonging to the eigenvalue λ ∈ C, i.e.

[x,y] = λ · y.

Because the subalgebra T ⊂ L is toral, the endomorphism


ad(y) : L−→ L

is semisimple. Again due to Lemma 5.2 also the endomorphism

adT (y) : T −→ T

is semisimple. Denote by (y j) j∈J a corresponding basis of eigenvectors of adT (y)belonging to eigenvalues (λ j) j∈J . In particular the element x ∈ T has a represen-tation

x = ∑j∈J

α j · y j.

We obtain

−λ · y =−[x,y] = [y,x] = adT (y)(x) = ∑j∈J

α j ·λ j · y j.

Assume λ 6= 0. Then for at least one j ∈ J

α j ·λ j 6= 0.

Hence the eigenvector y ∈ T of adT (y) with eigenvalue 0 is a linear combinationof eigenvectors belonging to eigenvalues λ j 6= 0, a contradiction. Therefore λ = 0,which implies

adT (x)(y) = 0, q.e.d.

5.2 sl(2,C)-modules

The 3-dimensional complex Lie algebra sl(2,C) is the prototype of complex semisim-ple Lie algebras. Also its representation theory is of fundamental importance:

• The representation theory of sl(2,C) is the means to clarify the structure of gen-eral complex semisimple Lie algebras, see Proposition 7.3 about the Cartan de-composition. Proposition 5.4 presents the structure of sl(2,C) in a form whichgeneralizes to arbitrary complex semisimple Lie algebras.

• The representation theory of sl(2,C) is paradigmatic for the representationtheory of general semisimple Lie algebras, see Chapter 8. Proposition 5.6,Corollary 5.7 and Theorem 5.8 present the representation theory of sl(2,C)-modulesin a form which generalizes to arbitrary complex semisimple Lie algebras.

Proposition 5.4 (Structure of sl(2,C)). The Lie algebra

L := sl(2,C)

has the standard basis B := (h,x,y) with matrices

5.2 sl(2,C)-modules 131

h =

(1 00 −1

),x =

(0 10 0

),y =

(0 01 0

)∈ sl(2,C).

• The non-zero commutators of the elements from B are

[h,x] = 2x, [h,y] =−2y, [x,y] = h.

• With respect to B the matrices of the adjoint representation

ad : L→ gl(L)

are

ad h =

0 0 00 2 00 0 −2

, ad x =

0 0 1−2 0 0

0 0 0

, ad y =

0 −1 00 0 02 0 0

• The element h ∈ L is ad-semisimple: ad h is a diagonal matrix. It has the eigen-

values0,α := 2,−α

with eigenspaces respectively

H := L0 = C ·h,Lα = C · x,L−α = C · y.

In particularL = H⊕L2⊕L−2

as a direct sum of complex vector spaces.

• The Lie algebra L is simple.

• With respect to the basis B the Killing form of L, see Definition 4.2, has thesymmetric matrix with integer coefficients

4 ·

2 0 00 0 10 1 0

∈M(3×3,Z).

The Killing form is non-degenerate.

• The Abelian subalgebra H ⊂ L is a maximal toral subalgebra of L.

• The restrictionκH := κ|H×H

of the Killing form is positive definite: κH(h,h) = 8. The scalar product κH in-duces the isomorphism

H ∼−→ H∗,h 7→ κH(h,−) = 8 ·h∗.


The element tα ∈ H with κH(tα ,−) = α ·h∗ = 2 ·h∗ is

tα =h4.

It satisfies

h =2 · tα

κ(tα , tα).

Proof. Only the following issues need a separate proof:

i) Simpleness: Due to the commutator relations L is not Abelian. Assume theexistence of a proper ideal I ⊂ L:

0( I ( L.

The splittingL = H⊕L2⊕L−2

implies the splitting of the ideal I

I = (I∩H)⊕ (I∩L2)⊕ (I∩L−2)

according to Lemma 5.2. We now use once more the fact [I,L]⊂ I.

• First, h /∈ I: Otherwise 2x = [h,x] ∈ I and −2y = [h,y] ∈ I, which implies I = L,a contradiction.

• Secondly, x /∈ I: Otherwise h = [x,y] ∈ I, contradicting the first part.

• Thirdly, y /∈ I: Otherwise −h = [y,x] ∈ I, contradicting the first part.

Each of the three summands in the splitting of I has dimension at most = 1.Hence I = 0, a contradiction, proving the non-existence of I. Therefore L issimple.

ii) To prove that H ⊂ sl(2,C) is a maximal toral subalgebra, we show CL(H) = H:

[α ·h+β · x+ γ · y,h] =−2β · x+2γ · y = 0 ⇐⇒ β = γ = 0 (α,β ,γ ∈ C).

Proposition 5.3 about the commutativity of toral subalgebras implies that H ismaximal toral, q.e.d.

The element

h ∈ L := sl(2,C)

is ad-semisimple by definition because ad h ∈ End(L) is semisimple. Therefore h ∈ Lcoincides with its semisimple component in the abstract Jordan decomposition of L.


Corollary 4.27 shows the far reaching consequences: The element h acts as semisim-ple endomorphism on any L-module V . Hence any L-module decomposes as a directsum of eigenspaces with respect to the action of h.

Definition 5.5 (Weight, weightspace and primitive element). Consider an sl(2,C)-module V ,not necessarily finite-dimensional.

1. Denote byV λ = v ∈V : h.v = λ ·v,λ ∈ C,

the eigenspaces with respect to the action of h. If

V λ 6= 0

then V λ is named a weight space of V , its non-zero elements are named weightvectors, and the corresponding eigenvalue λ ∈ C is a weight of V .

2. A weight vector e ∈ V λ is named a primitive element of V with weight λ

if x.e = 0 with respect to the action of the element x from the standard basis Bof sl(2,C).

For any sl(2,C)-module V not only the action of h ∈ sl(2,C), but also the actionof the two other elements of the standard basis

B = (h,x,y)

can be easily described.

Proposition 5.6 (Action of the standard basis). Consider an sl(2,C)-module V ,not necessarily finite-dimensional. Assume the existence of a primitive element e∈Vwith weight λ ∈ C. Then the elements

ei :=1i!· yi.e, i≥ 0, e−1 := 0,

satisfy for all i≥ 0:

1. Weight: h.ei = (λ −2i) · ei.

2. Lowering the weight: y.ei = (i+1) · ei+1.

3. Raising the weight: x.ei = (λ − i+1) · ei−1,e−1 := 0.

4. Linear dependence or independence:

• Either the family (ei)i≥0 is linearly independent, or


• the weight λ is a non-negative integer

λ =: m ∈ N,

the family (ei)i=0,...,m is linearly independent, and ei = 0 for all i > m.

Figure 5.1 illustrates the content of Proposition 5.6.

Fig. 5.1 Raising and lowering weights in irreducible sl(2,C)-modules

Proof. ad 2) From the definition

y.ei =1i!

y.(yi.e) =1i!(yi+1.ei+1) = (i+1) ei+1.

ad 1) Induction on i ∈ N and using part 2): i = 0 by definition. Inductionstep i 7→ i+1:

(i+1) (h.ei+1) = h.(y.ei) = [h,y].ei + y.(h.ei) =−2y.ei + y.((λ −2i)ei) =


= (λ −2i−2) (y.ei) = (λ −2i−2) (i+1)ei+1,

henceh.ei+1 = (λ −2i−2) ei+1.

ad 3) With the convention e−1 := 0 the formula

x.ei = (λ − i+1) · ei−1

follows by induction on i ∈ N by using the commutator formula

[x,y] = h ∈ sl(2,C)

together with the result from part 1) and part 2):

i · x.ei = x.(y.ei−1) = [x,y].ei−1 + y.(x.ei−1) = h.ei−1 + y.((λ − i+2) · ei−2) =

= (λ −2(i−1)) · ei−1 +(i−1)(λ − i+2) · ei−1 = i(λ − i+1) · ei−1

and after dividing by ix.ei = (λ − i+1) · ei−1.

Applying x ∈B raises the weight by 2.

ad 4) The non-zero elements ei have distinct weights. Hence they belong todifferent eigenvalues and are linearly independent. The family (ei)i≥0 is linearlyindependent when ei 6= 0 for all i ∈ N. Otherwise there exists a largest index m ∈ Nwith all elements e0, ...,em non-zero. Then ei = 0 for all i > m. We apply theformula from part 3) and obtain

0 = x.em+1 = (λ −m)em

which implies λ = m, q.e.d.

Corollary 5.7 (Primitive element). Consider a finite-dimensional sl(2,C)-module V .

1. Then V has a primitive element.

2. Any primitive element e ∈V has a non-negative integer weight m ∈ Z, m≥ 0. Itgenerates an irreducible sl(2,C)-submodule

V (m) := span < ei : i = 0, ....,m >⊂V.

Proof. 1. The subalgebra

B := spanC < h,x >⊂ sl(2,C)

is solvable. According to Lie’s theorem, see Theorem 3.20, the sl(2,C)-operationon V has a common eigenvector e ∈ V . Consider the eigenvalues λh, λx definedby


h.e = λh · e and x.e = λx · e.

The commutator [h,x] = 2x implies

[h,x].e = h.(x.e)− x.(h.e) = 0 and 2x.e = 2λx.e,

hence λx = 0. Therefore e ∈V is a primitive element with weight λh ∈ C.

2. Let e ∈ V be a primitive element with weight λ ∈ C. Because V is finite-dimensional, Proposition 5.6 implies:

λ =: m ∈ N.

and successive application of y lowers the weight up to the value−m. Therefore λ ≥ 0.

The vector space

W := spanC < ei : i = 0, ...,m >⊂V

is a submodule of V . All weight spaces W µ are 1-dimensional, generated bythe element ei with µ = m− 2i. Any non-zero submodule W ′ ⊂ W containsat least one weight vector, hence at least one element ei. The formulas fromProposition 5.6 imply that W ′ contains also e and a posteriori the whole sl(2,C)-module W .Therefore W ′ =W , which proves the irreducibility of W , q.e.d.

Chapter 8 will generalize Proposition 5.6 and Corollary 5.7 from sl(2,C) to ar-bitrary semisimple Lie algebras L. The solvable Lie algebra B will be called a Borelsubalgebra of L. The submodule generated by a primitive element will be named astandard cyclic module, cf. Definition 8.16.

The construction of the sl(2,C)-module V (λ ) from Proposition 5.6 generates allirreducible sl(2,C)-modules.

Theorem 5.8 (Classification of finite-dimensional irreducible modules).

1. For any λ ∈N a finite-dimensional, irreducible sl(2,C)-module V with primitiveelement e ∈V λ exists.

Any finite-dimensional, irreducible sl(2,C)-module V with primitive element e ∈V λ

is isomorphic to its submodule V (λ ). In particular, V splits as the direct sumof 1-dimensional weight spaces

V =λ⊕

i=0

V λ−2i.

All weights of V are integers.


2. The map to the isomorphy classes of finite-dimensional, irreducible sl(2,C)-modules

N→[V ] : V finite-dimensional, irreducible sl(2,C)-module,λ 7→ [V (λ )],

is bijective.

Proof. Set L := sl(2,C).

1. Choose a vector space V of dimension λ +1 with basis (e0, ...,eλ ). Define

L×V −→V

according to the formulas from Proposition 5.6:

• h.ei := (λ −2i) · ei• y.ei := (i+1) · ei+1, eλ+1 := 0• x.ei := (λ − i+1) · ei−1, e−1 := 0

and by linear extension. Due to the commutator relations of sl(2,C)

[h,x] = x, [h,y] =−2y, [x,y] = h

one has to check, that these definitions satisfy the equations

• h.(x.ei)− x.(h.ei) = 2x.ei• h.(y.ei)− y.(h.ei) =−2y.ei• x.(y.ei)− y.(x.ei) = h.ei

Accordingly, V becomes an L-module. It has the primitive element e0 ∈V λ , andis isomorphic to its irreducible submodule V (λ ) from Corollary 5.7.

2. According to part 1) the map from the theorem is well-defined.

Surjectivity: Consider a finite-dimensional irreducible sl(2,C)-module V , choosea primitive element e ∈V λ , and consider the submodule

V (λ )' span < yi.e : i ∈ N>⊂V.

The irreducibility of V implies V 'V (λ ).

Injectivity: If [V (λ1)] = [V (λ2)] then

V (λ1)'V (λ2),

in particular

1+λ1 = dim V (λ1) = dim V (λ2) = 1+λ2, q.e.d.


Corollary 5.9 (Classification of finite-dimensional modules). Any finite-dimensional sl(2,C)-module Vis a finite direct sum of irreducible modules of type V (λ )

V =⊕λ∈N

V (λ )nλ , nλ ∈ N,

with nλ 6= 0 for at most finitely many λ .

Proof. The proof follows from Weyl’s theorem on complete reducibility, see Theo-rem 4.24 and Theorem 5.8, q.e.d.

Example 5.10 (Explicit representation of finite-dimensional irreducible modules byhomogeneous polynomials).

Set L := sl(2,C).

1. The irreducible L-module of highest weight λ = 0 is the 1-dimensional vectorspace C with the trivial representation

ρ : sl(2,C)→0 ⊂ gl(C).

Its weight space decomposition is

V (0) =V 0 ' C.

Any non-zero element e ∈ C is a primitive element.

2. The irreducible L-module of highest weight λ = 1 is the 2-dimensional vectorspace C2 with the tautological representation

ρ : L = sl(2,C) −→ gl(C2).

Its weight space decomposition is

V (1) =V 1⊕V−1

with

V 1 = C ·(

10

),V−1 = C ·

(01

)and primitive element

e =(

10

).

3. The irreducible L-module of highest weight λ = 2 is the 3-dimensional vectorspace L itself, considered as L-module with respect to the adjoint representation

ad : L→ gl(L).


Proposition 5.4 shows the weight space decomposition

V (2) = L = L2⊕L0⊕L−2

with primitive element e = x ∈B.

4. In general, the irreducible L-module V (λ ) of highest weight λ = n ∈ N is iso-mophic to the complex vector space of all homogeneous polynomials

P(u,v) ∈ C[u,v]

of degree = n.

The vector space C[u,v] of polynomials in two variables u and v has a basisof monomials (uµ ·vν)µ,ν∈N. A homogeneous polynomial of degree n ∈ N is anelement

P(u,v) =n

∑µ=0

aµ ·uµ ·vn−µ ∈ C[u,v], aµ ∈ C.

Denote byPoln ⊂ C[u,v]

the subspace of homogeneoups polynomials of degree n. One has

dim Poln = n+1

because the family of monomials (un−i ·vi)i=0,...,n is a base of Poln.

When identifying the canonical basis of C2 with the two variables

u =

(10

)and v =

(01

)then the tautological representation of L acts on

Pol1 ' C ·u⊕C ·v

as the matrix product

z.(

ab

)= z ·

(ab

), z ∈ L.

We obtain

h.u = u, h.v =−v; x.u = 0, x.v = u; y.u = v, y.v = 0.

More general, we define an action of z ∈ L on Poln by the differential operator

Dz := (z.u)∂

∂u+(z.v)

∂

∂v,


notably for un−i ·vi ∈ Poln

h.(un−i ·vi) :=Dh(un−i ·vi)= u·(n−i)·un−i−1 ·vi−i ·v·un−i ·vi−1 =(n−2i)·un−i ·vi

y.(un−i ·vi) := Dy(un−i ·vi = v · (n− i) ·un−i−1 ·vi = (n− i) ·un−i−1 ·vi+1

x.(un−i ·vi) := Dx(un−i ·vi = u ·un−i · i ·vi−1 = i ·un−i+1 ·vi−1.

As a consequence, the map

L×Poln→ Poln,(z,P) 7→ Dz(P),

defines the irreducible L-module structure V (n) on Poln: We set

e := un ∈ Poln

and define for i = 0, ...,n

ei :=1i!· (yi.e) =

1i!·n · (n−1) · ... · (n− i+1) ·un−i · vi =

(ni

)·un−i · vi.

We obtainh.ei = (n−2i) · ei

y.ei = y.((

ni

)·un−i ·vi

)=

(ni

)· (n− i) ·un−i−1 ·vi+1 =

= (i+1) ·(

ni+1

)·un−i−1 ·vi+1 = (i+1) · ei+1

x.ei = x.((

ni

)·un−i ·vi

)= i ·

(ni

)·un−i+1 ·vi−1 =

(n− i+1) ·(

ni−1

)·un−i+1 ·vi−1 = (n− i+1) · ei−1,

which proves Poln 'V (n) with primitive element e := un ∈ Poln.

One checks that the isomorphy of vector spaces

Polλ = Symλ (C ·u⊕C ·v)' SymλC2

induces an isomorphy of L-modules between Polλ and the symmetric power withexponent λ of the tautological L-module.

5.3 Cartan subalgebra and root space decomposition

In this section the Lie algebra L is always assumed non-zero.

5.3 Cartan subalgebra and root space decomposition 141

For a semisimple Lie algebra L we choose an arbitrary, but fixed maximal toralsubalgebra T ⊂ L. According to Proposition 5.3 any toral subalgebra is Abelian.Hence all endomorphisms

ad h : L→ L, h ∈ T,

are simultaneously diagonizable, and the whole Lie algebra L splits as a direct sumof the common eigenspaces of T . This decomposition is called the root space de-composition of L with respect to T .

We recall from Lemma 4.1 that the Killing form of L is “associative”

κ([x,y],z) = κ(x, [y,z]),x,y,z ∈ L.

Definition 5.11 (Root space decomposition). Consider a pair (L,T ) with a semisim-ple Lie algebra L and a maximal toral subalgebra T ⊂ L.

1. For a complex linear functional

α : T → C

setLα := x ∈ L : [h,x] = α(h) · x f or all h ∈ T.

If α 6= 0 and Lα 6= 0 then α is a root of (L,T ) and Lα is the corresponding rootspace. Any non-zero vector v ∈V λ is named a root vector.

2. The set of all roots of (L,T ) is denoted

Φ := α : T → C : α 6= 0,Lα 6= 0.

3. The vector space decomposition

L = L0⊕⊕α∈Φ

Lα

is the root space decomposition of (L,T ).

In the root space decomposition the zero eigenspace L0 plays a distinguishedrole. We will show

T = L0.

By definition L0 =CL(T ). Proposition 5.3 shows that T is Abelian, hence T ⊂CL(T ).Therefore, the main step is the proof of the inclusion CL(T )⊂ T . See Theorem 5.16for the equality

T =CL(T ).

The main steps of the proof are:

• The centralizer CL(T ) contains with each element also its ad-semisimple andits ad-nilpotent summand. Therefore one can consider both types of elementsseparately.


• The case of ad-semisimple elements is trivial due to the maximality of T withrespect to ad-semisimple elements.

• If non-zero ad-nilpotent elements of CL(T ) exist, they cannot belong to T . Aposteriori, as a subset of T the centralizer CL(T ) cannot contain any non-zero ad-nilpotent elements because all elements of a toral subalgebra are ad-semisimple.

• Because the subalgebra CL(T ) is Abelian all its ad-nilpotent elements belong tothe null space of the Killing form restricted to CL(T ).

• The Killing form is nondegenerate on CL(T ).

The following Lemmata 5.12 and 5.14 as well as Proposition 5.15 prepare theproof of Theorem 5.16.

Lemma 5.12 (Orthogonality of root spaces). Consider a pair (L,T ) with a semisim-ple Lie algebra L and a maximal toral subalgebra T ⊂ L. Consider the correspond-ing root space decomposition from Definition 5.11

L = L0⊕α∈Φ

Lα .

For two functionals α,β ∈ T ∗:

•[Lα ,Lβ ]⊂ Lα+β .

• If β 6=−α thenκ(Lα ,Lβ ) = 0.

Proof. i) Assume x ∈ Lα ,y ∈ Lβ ,h ∈ T :

[h, [x,y]] =−([x, [y,h]]+ [y, [h,x]]) = [x,β (h) · y]− [y,α(h) · x]

= β (h) · [x,y]+α(h) · [x,y] = (α +β )(h) · [x,y] ∈ Lα+β .

ii) For the proof of the second claim choose an element h ∈ T with

(α +β )(h) 6= 0.

For arbitrary elements x ∈ Lα ,y ∈ Lβ :

κ([h,x],y) =−κ([x,h],y) =−κ(x, [h,y]),

α(h) ·κ(x,y) =−β (h) ·κ(x,y),

(α +β )(h) ·κ(x,y) = 0

which implies κ(x,y) = 0. Hence κ(Lα ,Lβ ) = 0, q.e.d.

Corollary 5.13 (Negative of a root). For a semisimple Lie algebra L the negative (−α)of a root α ∈Φ is again a root of L.


Proof. Assume on the contrary that −α is not a root. Then for all roots β ∈Φ

α +β 6= 0.

Lemma 5.12 implies

κ(Lα ,L0) = 0 and κ(Lα ,Lβ ) = 0.

Therefore the root space decomposition

L = L0⊕⊕β∈Φ

Lβ

impliesκ(Lα ,L) = 0.

Theorem 4.12 on the non-degeneratedness of the Killling form implies Lα = 0, acontradiction, q.e.d.

Lemma 5.14 (Centralizer of a maximal toral subalgebra). Consider a pair (L,T )with L a semisimple Lie algebra and T ⊂ L a maximal toral subalgebra.

i) The centralizer CL(T ) contains with each element x ∈ CL(T ) also the ad-semisimple and the ad-nilpotent part

s, n ∈CL(T )

from the abstract Jordan decomposition x = s+n.

ii) Any ad-semisimple element x ∈CL(T ) belongs to T .

Proof. Set C :=CL(T ).

i) If x ∈C with abstract Jordan decomposition x = s+ n then also s,n ∈C: Theabstract Jordan decomposition uses the fact that

ad x = ad s+ad n ∈ End(L)

is the Jordan decomposition of the endomorphism ad x ∈ End(L). In particular

ad s = ps(ad x) and ad n = pn(ad x)

with polynomials ps(T ), pn(T ) ∈ C[T ] satisfying ps(0) = pn(0) = 0.

As a consequence: If h ∈ T with (ad x)(h) = 0 then also

(ad s)(h) = 0 and (ad n)(h) = 0.


Hence s,n ∈C.

ii) All ad-semisimple elements x∈C belong to T : Any ad-semisimple element x ∈Ccommutes with all elements from T . Therefore ad x and all elements ad t, t ∈ T,are simultaneously diagonalizable. In particular, all elements from the vector space

ad(T +C · x)⊂ ad(L)

are diagonalizable. The maximality of T implies

T +C · x⊂ T,

i.e. x ∈ T , q.e.d.

The strategy to prove Theorem 5.16 uses the fact that the Killing form restrictedto a maximal toral subalgebra is non-degenerate. The following Proposition 5.15proves this result in two steps. Nevertheless, Theorem 5.16 will show a posteriorithat both steps coincide.

Proposition 5.15 (Non-degenerateness of the Killing form of a maximal toralsubalgebra). Consider a pair (L,T ) with L a semisimple Lie algebra and T ⊂ L amaximal toral subalgebra.

The restriction of the Killing form to the centralizer of T

κC := κ|CL(T )×CL(T ) : CL(T )×CL(T )→ C,

and the restriction to T itself

κT := κ|T ×T : T ×T → C

are non-degenerate.

Proof. 1) Restriction to CL(T ): By definition CL(T )=L0. Assume x∈L0 with κ(x,L0) = 0.According to Lemma 5.12: For any root β ∈Φ holds κ(x,Lβ ) = 0 . Hence

κ(x,L) = 0.

Because the Killing form is non-degenerate on L according to the Cartan criterionfrom Theorem 4.12, the last equation implies x = 0.

2) Restriction to T : Assume an element h ∈ T with κT (h,T ) = 0. Because T ⊂Cand because the restricted Killing form κC is nondegenerate according to the firstpart, it suffices to show

κC(h,C) = 0.

An ad-nilpotent element n ∈ C defines a nilpotent endomorphism ad n ∈ End(L).Because [x,h] = 0 also


[ad n,ad h] = 0.

Therefore the endomorphism

(ad h) (ad n) ∈ End(L)

is nilpotent. HenceκC(h,n) = tr((ad h) (ad n)) = 0

by Corollary 4.1. For a general element x ∈C with abstract Jordan decomposition

x = s+n

we first have s ∈ C due to Lemma 5.14, and then s ∈ T due to the same Lemma.Hence

κC(h,x) = κC(h,s)+κC(h,n) = κT (h,s)+κC(h,n) = 0

which implies h = 0, q.e.d.

Theorem 5.16 (A maximal toral subalgebra equals its centralizer). Consider asemisimple Lie algebra L and a maximal toral subalgebra T ⊂ L. Then

T =CL(T ).

Proof. According to Proposition 5.3 the toral subalgebra T is Abelian. Therefore T ⊂CL(T ).It remains to prove the opposite inclusion

CL(T )⊂ T.

For the proof set C := CL(T ). Due to Lemma 5.14 the main step to be proved willbe: Any ad-nilpotent element n ∈C belongs to T . For proving that step we will usethe non-degenerateness of the restricted Killing form, see Proposition 5.15.

i) T ∩ [C,C] = 0: By definition of the centralizer [C,T ] = 0. Lemma 4.1implies

0 = κC([T,C],C) = κC(T, [C,C]).

In particular,0 = κC(T,T ∩ [C,C]) = κT (T,T ∩ [C,C]).

Non-degenerateness of κT according to Proposition 5.15 implies T ∩ [C,C] = 0.

ii) The centralizer C is nilpotent: According to Engel’s theorem, see Theorem 3.10,it suffices to show that for each element x ∈C the endomorphism

adC x ∈ End(C)

is nilpotent. For the proof consider the abstract Jordan decomposition


x = s+n.

On one hand, Lemma 5.14 implies s ∈ T . Hence [s,C] = 0 by definition. On theother hand, the endomorphism ad n ∈ End(L) is nilpotent, hence a posteriori alsothe restriction (ad n)|C ∈ End(C). As a consequence,

adC x = adC n

is nilpotent.

iii) The centralizer C is Abelian: We argue by indirect proof. Assume on thecontrary [C,C] 6= 0. Due to part ii) the Lie algebra C is nilpotent. We applyCorollary 3.13 to the ideal

0 6= I := [C,C]⊂C

and obtain an element0 6= x ∈ Z(C)∩ [C,C].

The element x is not ad-semisimple, because ad-semisimple elements from C belongto T according to Lemma 5.14 and

T ∩ [C,C] = 0

according to part i). Therefore n 6= 0 in the abstract Jordan decomposition

x = s+n

and n∈C according to Lemma 5.14. Moreover x∈ Z(C) implies n∈ Z(C) accordingto Theorem 1.15. The nilpotency of ad n and the property [n,y] = 0 for all y ∈ Cimply the nilpotency of

(ad n) (ad y).

As a consequence κC(n,y) = 0 according to Lemma 4.1. We obtain κC(n,C) = 0.Proposition 5.15 implies n = 0, a contradiction.

iv) C⊂T : We argue by indirect proof. Assume the existence of an element x ∈C \Twith abstract Jordan decomposition

x = s+n.

First n 6= 0, because otherwise x = s is semisimple and Lemma 5.14 implies x ∈ T , acontradiction. Secondly, Lemma 5.14 implies n∈C. Thirdly, in order to show κC(n,C) = 0we consider an arbitrary element y ∈ C. The endomorphism adC(n) is nilpotent.Moreover [n,y] = 0 because C is Abelian by part iii). Hence the composition

adC(n)adC(y)

is nilpotent and


κC(n,y) = trace(adC(n)adC(y)) = 0.

Eventually, Proposition 5.15 implies n = 0, a contradiction, q.e.d.

Consider a pair (L,T ) with a complex semisimple Lie algebra L and a maximaltoral subalgebra T ∈ L. Theorem 5.16 allows to replace in the root space decom-position of L from Definition 5.11 the eigenspace L0 = CL(T ) by T . Moreover,Corollary 5.13 allows to choose from each pair (α,−α) of roots of L exactly onemember. The set of chosen members is denoted Φ+ ⊂Φ . How to make a canonicalchoice will be explained in Chapter 6 and 7.

Definition 5.17 (Cartan decomposition). Consider (L,T ) with a semisimple Liealgebra L and a maximal toral subalgebra T ⊂ L. Denote by Φ the root set of (L,T ).Then the decomposition as the direct sum of eigenspaces of T

L = T ⊕⊕α∈Φ

Lα = T ⊕⊕

α∈Φ+

(Lα ⊕L−α)

is the Cartan decomposition of L with respect to T .

Here we have applied Corollary 5.13: For any root α ∈Φ also (−α)∈Φ . Hencewe may choose from each pair (α,−α) of roots of L exactly one member. The setof chosen members is denoted by Φ+ ⊂Φ . How to make a canonical choice will beexplained in Chapter 6 and 7.

The Cartan decomposition of L with respect to T is the root space decompositionof L from Definition 5.11 with the additional information

T = L0

from Theorem 5.16. As a consequence, a complex semisimple Lie algebra L decom-poses as the direct sum of a maximal toral subalgebra T and the root spaces Lα ofits corresponding roots α ∈Φ .

Definition 5.18 (Cartan subalgebra). Consider a Lie algebra L. A Cartan subalge-bra H of L is a nilpotent subalgebra H ⊂ L equal to its normalizer, i.e. H = NL(H).

Lemma 5.19 (Cartan subalgebras of a semisimple Lie algebra). For a semisim-ple Lie algebra L any maximal toral subalgebra T ⊂ L is a Cartan subalgebra of L.


Proof. i) According to Proposition 5.3 any toral subalgebra T ⊂ L is Abelian, inparticular nilpotent.

ii) The Cartan decomposition of L with respect to T represents an arbitraryelement x ∈ NL(T ) uniquely as

x = xT + ∑α∈Φ

xα , xT ∈ T, xα ∈ Lα .

For all h ∈ T

[h,x] = [h,xT ]+ ∑α∈Φ

α(h) · xα = ∑α∈Φ

α(h) · xα ∈ T.

For any α ∈Φ an element h ∈ T exists with α(h) 6= 0. Then [h,x] ∈ T implies

xα = 0.

We obtain x = xT . Therefore NL(T )⊂ T . Apparently T ⊂ NL(T ). As a consequence

NL(T ) = T, q.e.d.

Remark 5.20 (Cartan subalgebras of a semisimpleLie algebra). For a semisimpleLie algebra L the two concepts Cartan subalgebra and maximal toral subalgebraare even equivalent, see [28, Chapter 15.3].

In the following we will employ the standard notation H for Cartan subalgebras,and denote by the letter H a fixed toral subalgebra of a semisimple Lie algebra L,i.e. a fixed Cartan subalgebra of L.

Chapter 6Root systems

Our point of departure is the Cartan decomposition of a complex semisimple Liealgebra

L = H⊕⊕α∈Φ

Lα ,

see Definition 5.17.We separate the concept of roots from the Lie algebra as its origin and study

the properties of Φ in the context of abstract root systems. Here we follow Serre’sguide [43]. Different from many other authors Serre introduces a root system in anaxiomatic way by its set of reflections. The existence of an invariant scalar productis then a consequence and not a prerequisite. Serre develops the properties of a rootsystem by focusing on the real vector space V spanned by the root system.

The base field in the present chapter is R, all vector spaces are real and finitedimensional.

6.1 Abstract root system

The ambient space of an abstract root system is a real vector space V . One may con-ceive of V as the vector space spanned by the root set of a semisimple Lie algebra L.The root set is a distinguished set of linear functionals on the Cartan algebra H ⊂ L.When following this intuition roots are conceived as linear functionals.

Definition 6.1 (Symmetry). Consider a vector space V . A symmetry of V with vec-tor α ∈V,α 6= 0, is an automorphism

σα : V →V

with

149

150 6 Root systems

1. σα(α) =−α

2. The fixed spaceHα := x ∈V : σα(x) = x

of elements fixed by σα is a hyperplane in V , i.e. codimV Hα = 1.

Apparently the hyperplane Hα is a complement of the real line R ·α ⊂ V . Andthe symmetry σα is completely determined by the choice of α and Hα .

The symmetry defines a linear functional α∗ ∈V ∗ such that for all x ∈V

σα(x) = x−α∗(x) ·α :

Ifx = µ1 ·α +µ2 ·v,v ∈ Hα ,

thenσα(x) =−µ1 ·α +µ2 ·v = x−2µ1 ·α.

Henceα∗(x) =−2 ·µ1.

The linear functional α∗ satisfies α∗(α) = 2 and ker α∗ = Hα .

Conversely, if 0 6= α ∈ V and a linear functional α∗ ∈ V ∗ with α∗(α) = 2 aregiven, then the definition

σα(x) := x−α∗(x) ·α, x ∈V,

defines a symmetry σα with fixed hyperplane Hα := ker α∗.

In the following we define a root system Φ . It has the decisive property that forall α ∈Φ the linear functional α∗ ∈V ∗ takes on integral values on all elements β ∈Φ .

Definition 6.2 (Root system). A root system of a vector space V is a subset Φ ⊂Vwith the following properties:

• (R1) Finite and generating: The set Φ is finite, 0 /∈Φ , and spanRΦ =V .

• (R2) Invariance under distinguished symmetries: For each α ∈Φ a symmetry σα

of V with vector α exists which leaves Φ invariant, i.e. σα(Φ)⊂Φ .

6.1 Abstract root system 151

• (R3) Cartan integers: For all α,β ∈Φ , integer values < β ,α >∈ Z exist with

σα(β ) = β−< β ,α > ·α.

The integers< β ,α >∈ Z,α,β ∈Φ ,

are named the Cartan integers of Φ .

• (R4) Reducedness: For each α ∈ Φ the only roots proportional to α are α itselfand −α , i.e.

(R ·α)∩Φ = α,−α.

The dimension of V is the rank of the root system, the elements of Φ are the rootsof V .

In the literature condition (R4) is considered an additional requirement for a re-duced root system. If the base field is not algebraically closed one has to distinguishbetween reduced and non-reduced root systems. But in these notes we only considerreduced root systems. Therefore we omit the attribute reduced.

Note that the Cartan integers < β ,α > are linear in the first argument β but notnecessarily in the second argument α . For any α ∈Φ :

< α,α >= 2.

Lemma 6.3 (Uniqueness of the symmetry). The symmetry σα required in Defini-tion 6.2 part (R2) is uniquely determined by α .

Proof. Assume two symmetries of V with vector α satisfying

σi(Φ)⊂Φ , i = 1,2.

Consider the automorphism of V

u := σ2 σ1.

It satisfiesu(α) = α and u(Φ) = Φ .

Moreover, u induces the identity on the quotient V/Rα because

u(x) = σ2(σ1(x)) = σ2(x−α∗1 (x) ·α) = x−α

∗2 (x) ·α−α

∗1 (x) ·α +α

∗2 (α

∗1 (x) ·α) ·α

impliesu(x) ∈ x+Rα.

As a consequence, u has the single eigenvalue λ = 1. Hence the characteristic poly-nomial pchar(T ) of u has the only root λ = 1. The minimal polynomial pmin(T )

152 6 Root systems

of u divides the characteristic polynomial pchar(T ) of u, see Theorem 1.16. Hencealso pmin(T ) has only the root λ = 1. Because u|Φ is a permutation of Φ , the re-striction has a finite order, i.e. an exponent n ∈ N exists with

(u|Φ)n = id|Φ .

Because Φ spans V we obtain un = id, i.e. the polynomial T n− 1 annihilates u.Therefore pmin(T ) divides the polynomial T n− 1, which has the root λ = 1 withalgebraic multiplicity one. As a consequence

pmin(T ) = T −1

and u = id, q.e.d.

Definition 6.4 (Weyl group). The Weyl group W of a root system Φ of V is thesubgroup of GL(V ) generated by all symmetries σα ,α ∈Φ .

The Weyl group permutes the elements of the finite set Φ , hence W is a finitegroup. As a consequence, one may average an arbitrary scalar product over the ele-ments of the Weyl group, obtaining an invariant scalar product with respect to W .

Lemma 6.5 (Invariant scalar product). Let Φ be a root system of a vector space V .Then a scalar product (−,−) exists on V which is invariant under the Weyl group Wof Φ . With respect to any invariant scalar product the Cartan integers satisfy

< β ,α >= 2 ·(β ,α)

(α,α), α,β ∈Φ .

Proof. Take an arbitrary scalar product B on V and define the bilinear form (−,−)as

(x,y) := ∑w∈W

B(w(x),w(y)), x,y ∈V,

as the average over the Weyl group, which is a finite group. Apparently, (−,−)is positive definite, hence a scalar product. By construction, the scalar productis W -invariant. For two roots α,β ∈ Φ the corresponding Cartan integer < β ,α >is defined by the equation

σα(β ) = β−< β ,α > α.

Becauseσα = σ

−1α ,σα(α) =−α,

and due to the invariance of the scalar product:

−(α,β ) = (σα(α),β ) = (α,σα(β )) = (α,β )−< β ,α > (α,α),


which proves

< β ,α >= 2 ·(α,β )

(α,α)= 2 ·

(β ,α)

(α,α), q.e.d.

Now that we have an Euclidean space (V,(−,−)) of the root system Φ , we candefine the length of a root and the angle between two roots. We choose a fixedinvariant scalar product.

Definition 6.6 (Length of roots and angle between roots). Consider a root system Φ

and the corresponding Euclidean vector space (V,(−,−)).

1. The length of a root α ∈Φ is defined as

‖α‖ :=√

(α,α).

2. The angle0≤ θ := ^(α,β )≤ π

included between two roots α,β ∈Φ is defined as

cos(θ) :=(α,β )

‖α‖ · ‖β‖.

Lemma 6.7 (Possible angles and length ratio of two roots). Consider a root sys-tem Φ and two non proportional roots α,β ∈Φ . Table 6.1 displays the only possibleangles ^(α,β ) included by α and β and their ratios of length.

The last column of the table indicates the type of the root system of rank = 2 withbase ∆ = α,β, see Definition 6.10 and Theorem 6.26.

No. < α,β > < β ,α > ^(α,β ) ‖β‖2/‖α‖2 ∆ = α,β1 0 0 π

2 undef. A1×A1

2 1 1 π

3 1

3 -1 -1 2π

3 1 A2

4 1 2 π

4 2

5 -1 -2 3π

4 2 B2

6 1 3 π

6 3

7 -1 -3 5π

6 3 G2

Table 6.1 Angles and length of roots

154 6 Root systems

Proof. Employing the Cartan integers < α,β >,< β ,α > ∈ Z we have

4 > 4 · cos2θ = 2 · (α,β )

‖β‖2 ·2 ·(β ,α)

‖α‖2 = < α,β > ·< β ,α > ≥ 0.

Both Cartan integers have equal sign and we may assume ‖β‖ ≥ ‖α‖, i.e.

|< α,β > | ≤ |< β ,α > |

because‖β‖2

‖α‖2 =< β ,α >

< α,β >=|< β ,α > ||< α,β > |

, q.e.d.

The formulas from the proof of Lemma 6.7 indicate : For two non-propotionalroots α,β ∈Φ the product of their Cartan integers determines the included angle θ = ^(α,β ),while the quotient of the Cartan integers determines their length ratio ‖β‖/‖α‖.Moreover, Table 6.1 shows that the angle determines the length ratio with the onlyexception of the orthogonal case θ = π/2.

Example 6.8 (Root systems of rank ≤ 3).

• Rank = 1: The only root system is A1.

• Rank = 2: Figure 6.1 displays all root systems of rank = 2, see also Theorem 6.26.

• Rank = 3: See the figures in [22, Chap. 8.9] as one example.


Fig. 6.1 Root systems of rank = 2

Lemma 6.9 (Roots with acute angle). If two non-proportional roots α,β ∈ Φ in-clude an acute angle, i.e. (α,β )> 0, then also α−β ∈Φ .

Proof. According to Table 6.1 the assumption (α,β )> 0 implies

< β ,α >= 1 or < α,β >= 1.

In the first case σα(β ) = β − α , in the second case σβ (α) = α − β . In bothcases α−β ∈Φ because the Weyl group leaves Φ invariant, q.e.d.

Definition 6.10 (Base of a root system, positive and negative roots). Consider aroot system Φ of a vector space V .

i) A set ∆ = α1, ...,αr of roots αi ∈ Φ , i = 1, ...,r, is a base of Φ and theelements of ∆ are named simple roots iff

• The family (αi)i=1,...,r is a basis of V .

156 6 Root systems

• Each root β ∈Φ has a representation

β =r

∑i=1

ki ·αi, ki ∈ Z,

with either all integer coefficients ki ≥ 0 or all all ki ≤ 0.

ii) With respect to a base ∆ a root β is

• positive, β 0, iff all ki ≥ 0• negative, β ≺ 0, iff all ki ≤ 0.

The subset Φ+ ⊂ Φ is defined as the set of all positive roots and the subsetsubset Φ− ⊂Φ as the set of all negative roots.

Theorem 6.11 (Existence of a base). Every root system Φ of a vector space V hasa base ∆ .

Proof. The construction of a candidate for ∆ is straightforward. But the proof thatthe candidate is indeed a base, will take several steps.

i) Construction of ∆ : Because Φ is finite, a linear functional t ∈ V ∗ existswith t(α) 6= 0 for all α ∈Φ . Set

Φ+t := α ∈Φ : t(α)> 0

and call α ∈Φ+ decomposable iff

α = α1 +α2 with α1,α2 ∈Φ+t

and indecomposable otherwise. We claim that the set

∆ := α ⊂Φ+t : α indecomposable

is a base of Φ .

ii) Representation of elements from Φ+t : Each β ∈Φ

+t has the form

β = ∑α∈∆

kα ·α

with all kα ≥ 0: Otherwise consider the non-empty subset C ⊂ Φ+t of all elements

which lack such a representation.Choose an element β ∈C with t(β ) > 0 minimal. By construction β is decom-

posable. Henceβ = β1 +β2

with β1,β2 ∈Φ+ and β1 ∈C or β2 ∈C. We get


t(β ) = t(β1)+ t(β2)

which implies0 < t(β1), t(β2)< t(β ),

a contradiction to the minimality of β .

iii) Angle between elements from ∆ : We claim that two different roots α 6= β

from ∆ are either orthogonal or include an obtuse angle, i.e. (α,β )≤ 0.Otherwise (α,β )> 0 and we obtain from Lemma 6.9 the root

γ := α−β ∈Φ .

As a consequence α = β +γ and γ /∈Φ+t because α indecomposable. Hence −γ ∈Φ

+t

which impliesβ = α +(−γ)

decomposable. The contradiction proves the claim.

iv) Linear independency: A finite subset A⊂V with all elements α,β ∈ A satis-fying

t(α)> 0 and (α,β )≤ 0.

is linearly independent.Assume the existence of a representation

0 = ∑α∈A

nα ·α

with coefficients nα ∈ R for all α ∈ A. Separating summands with positive coeffi-cients from those with negative coefficients gives an equation

∑α∈A1

kα ·α = ∑α∈A2

kα ·α =: v ∈V

with disjoint subsets A1,A2 ⊂ A and all kα ≥ 0. Then

(v,v) = ∑α∈A1,β∈A2

kα · kβ · (α,β )≤ 0.

Hence v = 0. Now0 = t(v) = ∑

α∈A1

kα · t(α)

with t(α) > 0 for all α ∈ A1 implies kα = 0 for all α ∈ A1. Similarly kα = 0 forall α ∈ A2.

The sequence of all steps i) until iv) proves the claim of the theorem, q.e.d.

158 6 Root systems

The proof of Theorem 6.11 constructs a base ∆ by starting from a certain func-tional t ∈V ∗. We show that any base can be obtained in this way: Any base of a rootsystem Φ is the set of indecomposable elements of Φ from the positive half-spaceof a suitable linear functional.

Lemma 6.12 (Base and a determining linear functional). Consider a base ∆ of aroot system Φ of a vector space V .

Then a functional t ∈V ∗ exists with t(α) 6= 0 for all α ∈Φ such that

Φ+ = Φ

+t := α ∈Φ : t(α)> 0

and∆ = α ∈Φ

+t : α indecomposable.

Proof. The base ∆ defines the decomposition

Φ = Φ+∪Φ

−.

Because (α)α∈∆ is a basis of the vector space V , a functional t ∈V ∗ exists with t(α)> 0for all α ∈ ∆ . Set

Φ+t := α ∈Φ : t(α)> 0,Φ−t := α ∈Φ : t(α)< 0.

From∆ ⊂Φ

+t

followsΦ

+ ⊂Φ+t and Φ

− ⊂Φ−t .

And the decompositionΦ

+∪Φ− = Φ = Φ

+t ∪Φ

−t

impliesΦ

+ = Φ+t and Φ

− = Φ−t .

As a consequence, the indecomposable elements from both sets Φ+ and Φ+t are

equal, i.e. ∆ = ∆ ′, q.e.d.

Corollary 6.13 (Simple roots include an obtuse angle). Consider a base ∆ of aroot system Φ of a vector space V .

Any two different roots α 6= β ∈ ∆ include an obtuse angle or are orthogonal,i.e. (α,β )≤ 0.

Proof. According to Lemma 6.12 a suitable functional t ∈V ∗ exists with

∆ = α ∈Φ : t(α)> 0 and α indecomposable.

Part iii) in the proof of Theorem 6.11 shows (α,β )≤ 0 because all elements from ∆

belong to the same half-space.

6.2 Action of the Weyl group 159

6.2 Action of the Weyl group

Our aim in the present chapter is the classification of all possible root systems. Thisresult derives from two facts.

The first is the integrality of the Cartan integers of a root system Φ of a realvector space V . This fact restricts the angle and the relative length of two roots, seeLemma 6.7. The second fact are the different group operations of the Weyl group, inparticular the transitive operation on the set of bases of a given root system. Besidesthe action of the Weyl group W on Φ

W ×Φ →Φ ,(w,α) 7→ w(α),

we study the action of W on

• the linear functionals defined by the Cartan integers < α,β >• on the symmetries σα : V −→V , and• on the bases ∆ of Φ .

We will show: A root system is characterized by the matrix of its Cartan integers,the Cartan matrix. Only finitely many types of Cartan matrices exist.

Definition 6.14 (Group action). A group G acts on a set X if a map exists

G×X −→ X , (g,x) 7→ g.x

with the following properties:

e.x = x for all x ∈ X and the neutral element e ∈ G

and(g1 ·g2).x = g1.(g2.x) for all g1,g2 ∈ G and all x ∈ X .

The group acts transitive if for all x ∈ X the induced map

G−→ X , g 7→ g.x,

is surjective.

Lemma 6.15 (Action of the Weyl group on Cartan integers and on symmetries).Consider a root system Φ of a vector space V with Weyl group W .

1. The Cartan integers are linear in the first argument, i.e. for all α ∈Φ the Cartanfunctional

<−,α >: V −→ C

is C-linear.

160 6 Root systems

2. Denote byA := <−,α >∈V ∗ : α ∈Φ

the set of all Cartan functionals. The map

W ×A−→ A, (w,<−,α >) 7→<−,α > w−1,

is a group operation.

3. Denote byB := sα : V −→V |α ∈Φ

the set of all symmetries. The map

W ×B−→ B, (w,σα) 7→ wσα w−1,

is a group action.

4. Denote byC := ∆ ⊂Φ : ∆ a base of Φ

the set of all bases of Φ . The map

W ×C −→C, (w,∆) 7→ w(α) : α ∈ ∆,

is a group action.

Proof. 1. According to Lemma 6.5 a scalar product (−,−) on V exists which isinvariant with respect to W . The formula

<−,α >= 2 ·(−,α)

(α,α), α ∈Φ ,

shows the C-linearity of the Cartan functional.

2. For w ∈W and α ∈Φ consider the root β := w(α) ∈Φ . Then

<−,α > w−1 =<−,β >:

For the proof compute

<−,β >= 2 ·(−,β )(β ,β )

= 2 ·(−,w(α))

(β ,β )= 2 ·

(w−1(−),α)

(β ,β )= <−,α > w−1.

Hence the map W ×A −→ A is well-defined. One checks that it satisfies the twoproperties of a group action.

3. For w ∈W and α ∈Φ consider the root β := w(α) ∈Φ . Then

sβ = wσα w−1 :


For any v ∈V one computes the left-hand side

σβ (v) = v−< v,β > ·β = (ww−1)(v)−< w−1(v),α > ·w(α) =

= w(w−1(v)−< w−1(v),α > ·α

).

And the right-hand side

(wσα w−1)(v) = w(σα(w−1(v))) = w(w−1(v)−< w−1(v),α > ·α).

Hence the map W ×B −→ B is well-defined. One checks that it satisfies the twoproperties of a group action.

4. The proof is obvious, q.e.d.

In accordance with Lemma 6.15 about the action on the Cartan integers one definesthe action of W on the dual space V ∗:

W ×V ∗→V ∗,(w, t) 7→ w(t) := t w−1.

Proposition 6.16 (Properties of the Weyl group). Consider a root system Φ of avector space V and denote by W its Weyl group. Denote by ∆ a fixed base of Φ .Then

1. Embedding ∆ into half-spaces: For any functional t ∈ V ∗ an element w ∈ Wexists with

t(w(α))≥ 0

for all α ∈ ∆ .

2. Transitive action on bases: The action of W on the set of bases of Φ is transitive,i.e. for any base ∆ ′ of Φ an element w ∈W exists with

w(∆) = ∆′.

3. Any root extends to a base: For any root α ∈Φ an element w ∈W exists with

w(α) ∈ ∆ .

4. Symmetries of roots from ∆ are generators: The Weyl group W is generated bythe symmetries σα of the roots α ∈ ∆ .

Proof. Denote by W∆ ⊂ W the subgroup generated by the symmetries σα of theroots α ∈ ∆ .

162 6 Root systems

i) The symmetry σα of any root α ∈∆ leaves the set Φ+ \α invariant: Assumethat ∆ comprises at least two roots. Consider an element β ∈Φ+ \α. It has arepresentation

β = ∑γ∈∆

kγ · γ,kγ ≥ 0, for all γ ∈ ∆ .

Because β is not proportional to α we have kγ > 0 for at least one γ ∈ ∆ \α. Weget

σα(β ) = β−< β ,α > α =

= (∑γ∈∆

kγ · γ)−< β ,α > α = (kα−< β ,α >) ·α + ∑γ∈∆\α

kγ · γ.

Hence also σα(β ) is not proportional to α and has at least one coefficient kγ > 0.According to Definition 6.10, part i), all coefficients are non-negative and

σα(β ) ∈Φ+ \α.

ii) Consider the distinguished element ρ ∈V defined as half the sum of all posi-tive roots

ρ := (1/2) · ∑β∈Φ+

β .

According to part i) each symmetry σα ,α ∈ ∆ , permutes all positive roots differentfrom α and σα(α) =−α . Hence

σα(ρ) = ρ−α.

We now show that the first three claims can be satisfied already by taking ele-ments from W∆ .

iii) Part 1) of the Proposition is satisfied even with an element w ∈ W∆ : For agiven functional t ∈ V ∗ we choose an element w ∈W∆ with t(w(ρ)) maximal withrespect to w ∈W∆ . For arbitrary α ∈ ∆ holds according to part ii):

α = ρ−σα(ρ)

w(α) = w(ρ)−w(σα(ρ))

t(w(α)) = t(w(ρ))− t(w(σα(ρ))).

Because also wσα ∈W∆ we have

t(w(ρ))≥ t(w(σα(ρ)))

ort(w(α))≥ 0.


iv) Also part 2) of the proposition is satisfied even with an element w ∈W∆ : First,Lemma 6.12 characterizes the base ∆ ′ by a functional t ′ ∈V ∗ with

t ′(α ′)> 0

for all α ∈ ∆ ′. Secondly, part 1) of the proposition transforms t ′ by a suitableelement w ∈W∆ to the functional

t := t ′ w ∈V ∗

such that for all α ∈ ∆

t(α)≥ 0.

The case t(α) = 0 is excluded, because otherwise the root w(α) ∈Φ would satisfy

t ′(w(α)) = 0,

which is excluded because t ′|∆ ′ > 0. Applying again Lemma 6.12 characterizes ∆

as the set of indecomposable roots in the positive half-space of the functional t.Therefore

∆ = w(∆ ′) := w(α ′) : α′ ∈ ∆

′.

v) Also part 3) of the Lemma is satisfied even with an element w ∈W∆ : Forfixed α ∈Φ we find an element t0 ∈V ∗ with t0(α) = 0 but t0(β ) 6= 0 for all roots β

not proportional to α . Because Φ is a finite set, the minimum

min|t0(β )| : β ∈Φ not proportional to α

is positive. Hence a small perturbation of t0 provides a linear functional t ∈V ∗ andan ε > 0 such that for all roots β not proportional to α

|t(β )|> ε

butt(α) = ε.

Denote by ∆(t) the base of Φ induced by t according to Lemma 6.12. By part iv)an element w ∈W∆ exists with

w(∆(t)) = ∆ .

The root α ∈ Φ is contained in the positive half-space Φ+t of the functional t. It is

indecomposable because t(β )> ε for all roots β ∈Φ+t which are non-proportional

to α . Therefore α ∈ ∆(t), which implies w(α) ∈ ∆ .

vi) We prove W∆ =W : Because the Weyl group is generated by all symmetries σα , α ∈Φ ,it suffices to show that

σα ∈W∆

164 6 Root systems

for all α ∈ Φ . We choose an arbitrary root α ∈ Φ . According to part iv) anelement w ∈W∆ exists with

β := w(α) ∈ ∆ .

Thenσβ = σw(α) = wσα w−1

σα = w−1 σβ w ∈W∆ , q.e.d.

Definition 6.17 (Cartan matrix). Consider a root system Φ of a vector space V anda base ∆ = α1, ...,αr of Φ .

The Cartan matrix of ∆ is the matrix of the Cartan integers of the roots from ∆

Cartan(∆) := (< αi,α j >1≤i, j≤r) ∈M(r× r,Z).

Here the index i denotes the row and the index j denotes the column.

Note that the Cartan matrix is not necessarily symmetric. All diagonal elementsof the Cartan matrix have the value

< αi,αi >= 2.

For i 6= j only values< αi,α j >≤ 0

are possible according to Corollary 6.13. Moreover, these values are restricted to theset

0,−1,−2,−3

according to Lemma 6.7 and Corollary 6.13.

The Cartan matrix is defined with reference to a base ∆ and with reference toa numbering of its elements. Lemma 6.18 shows that for any two bases of Φ anynumbering of the elements of the first base induces a numbering of the elements ofthe second base such that the respective Cartan matrices are equal.

Lemma 6.18 (Independence of the Cartan matrix from the choosen base). Con-sider a root system Φ of rank r = rank Φ . Any two bases ∆ ,∆ ′ of Φ have the sameCartan matrix. More specifically:

An element w ∈W of the Weyl group of Φ exists with w(∆) = ∆ ′ and

Cartan(∆) = (< αi,α j >1≤i, j≤r) =Cartan(∆ ′) = (< w(αi),w(α j)>1≤i, j≤r).


Proof. According to Proposition 6.16 an element w∈W exists with w(∆) = ∆ ′, i.e.

∆ = α1, ...,αr =⇒ ∆′ = w(α1), ...,w(αr).

According to Lemma 6.15

<−,w(α j)>=<−,α j > w−1

which implies

< w(αi),w(α j)>=< w−1(w(αi)),α j >=< αi,α j >,q.e.d.

The Cartan matrix encodes the full information of the root system Φ , notably thedimension of its ambient space V :

Theorem 6.19 (The Cartan matrix characterizes the root system). Consider aroot system Φ of an Euclidean space V and a base

∆ = α1, ...,αr

of Φ . Let V ′ be a second Euclidean space and

∆′ = α ′1, ...,α ′r

a base of a root system Φ ′ of V ′. If a bijective map

f : ∆ → ∆′

exists with

Cartan(< αi,α j >1≤i, j≤r) =Cartan(< f (αi), f (α j)>1≤i, j≤r),

then a unique isomorphism of vector spaces

F : V →V ′

exists withF(Φ) = Φ

′ and F |∆ = f .

Proof. i) Determination of F : Because the elements from ∆ form a basis of thevector space V we may define

F : V →V ′

as the uniquely determined linear extension of f . And because ∆ and ∆ ′ have thesame cardinality the linear map F is an isomorphism.

ii) We show that the Weyl groups W and W ′ are conjugate via F , i.e.

W ′ = F W F−1 or W ′ F = F W :

166 6 Root systems

Consider two roots α,β ∈ ∆ and the corresponding symmetries. Then

(σ f (α) F)(β ) = σ f (α)( f (β )) = f (β )−< f (β ), f (α)> f (α)

(F σα)(β ) = F(σα(β )) = f (β−< β ,α,> α) = f (β )−< β ,α,> f (α)

Hence for every root α ∈ ∆

σ f (α) F = F σα .

Moreover, for two roots α1,α2 ∈ ∆ :

(σ f (α2) σ f (α1)F = σ f (α2) (σ f (α1) F) = σ f (α2) (F σα1) =

= (σ f (α2) F)σα1 = (F σα2)σα1 = F (σα2 σα1).

The Weyl groups W and W ′ are generated by the symmetries of the elements fromrespectively ∆ and ∆ ′, see Proposition 6.16. Hence

W ′ F = F W .

iii) We show F(Φ)⊂Φ ′: Consider an arbitrary root β ∈Φ . According to Proposi-tion 6.16 elements α ∈ ∆ and w ∈W exist with

β = w(α).

Due to part ii) an element w′ ∈W ′ exists such

F(β ) = (F w)(α) = (w′ F)(α) = w′( f (α)) ∈ w′(∆ ′)⊂Φ′, q.e.d.

6.3 Coxeter graph and Dynkin diagram

Due to Theorem 6.19 a root system Φ is completely determined by its Cartan matrixwith respect to a base ∆ of Φ . Hence the classification of root systems reduces to theclassification of possible Cartan matrices. Thanks to Lemma 6.7 and Corollary 6.13the Cartan integers satisfy a set of restrictions. The present section shows that thelanguage of Cartan matrices translates to a data structure from Discrete Mathemat-ics. The data structure is a weighted graph called the Coxeter graph of the rootsystem.

Therefore, we first define the Coxeter graph of ∆ , and then classify all possibleCoxeter graphs. This classification is achieved by a second language translation: Wetranslate the language of Coxeter graphs to the language of Euclidean vector spaces,and apply same basic results from Linear Algebra. The result is Theorem 6.21.

6.3 Coxeter graph and Dynkin diagram 167

Recall: For a base ∆ of Φ the angle θ between two distinct roots α 6= β ∈ ∆ isdetermined by the product of their Cartan integers as

< α,β >< β ,α >= 4 · cos2θ ∈ 0,1,2,3, π/2≤ θ < π.

If (−,−) denotes a scalar product on V which is invariant under Φ , then

< α,β >< β ,α >= 4 ·(α,β )2

‖α‖2 · ‖β‖2.

Definition 6.20 (Coxeter graph). Consider a root system Φ of a vector space V anda base ∆ of Φ . The Coxeter graph of Φ with repect to ∆ is the undirected weightedgraph

Coxeter(Φ) = (N,E)

with

• vertex set N := ∆

• and set E of weighted edges: A weighted edge (e,m) ∈ E is a pair with anedge e = α,β joining two vertices in N iff

α 6= β and < α,β >< β ,α >6= 0.

The corresponding weight is defined as

m :=< α,β >< β ,α >∈ 1,2,3.

Theorem 6.21 (Classification of connected Coxeter graphs). A connected Cox-eter graph of a root system Φ with respect to a base ∆ belongs to exactly one of theclasses from Figure 6.2 - for a suitable numbering of the elements of ∆ :

Explication:

• Series Ar,r ≥ 1: Each pair of subsequent roots include the angle 2π

3 .

• Series Br,r ≥ 2 or Series Cr,r ≥ 3: The first r− 2 pairs of subsequent rootsinclude the angle 2π

3 , the last two roots include the angle 3π

4 .

• Series Dr,r ≥ 4: The first r− 3 subsequent pairs of roots include the angle 2π

3 ,root αr−2 includes with each of the two roots αr−1 and αr the angle 2π

3 .

• Exceptional type G2: ∆ = (α1,α2). The two roots include the angle 5π

6 .

• Exceptional type F4: ∆ = (α1, ...,α4). The pairs (α1,α2) and (α3,α4) includethe angle 2π

3 , the pair (α2,α3) includes the angle 3π

4 .

168 6 Root systems

Fig. 6.2 Connected Coxeter graphs (annotation for weight = 1 suppressed)

• Exceptional series Er,r ∈ 6,7,8 : ∆ = (α1, ...,αr). The first r−2 pairs of sub-sequent roots include the angle 2π

3 . In addition, the distinguished root α3 includesthe angle 2π

3 with the root αr.

The range of r ∈ N for the types Ar,Br,Cr,Dr has been choosen to avoid dubletswith low value of r.

Note. The Coxeter graph employing the product of Cartan integers as its weightsdoes not encode the relative length of two roots, i.e. their length ratio. The lengthratio derives from the quotient of the Cartan integers(

‖β‖‖α‖

)2

=< β ,α >

< α,β >.

Knowing the product of the Cartan integers is not sufficient to reconstruct the Car-tan matrix. Hence the Coxeter graph does not encode the full information aboutthe root system. We will see that a base of the root systems belonging to thetypes Br,Cr,G2,F4 is made up by roots with different lengths. As a consequence,after complementing the Coxeter graph by the information about the length ratio theseries Br and Cr of root systems will differ for r ≥ 3, see Theorem 6.26.


But in any case, the (absolute) length of a root is not defined because an invariantscalar product of a root system is not uniquely determined.

For the proof of Theorem 6.21 we will capture the characteristic properties ofthe Coxeter graph in the following definition which embeds the graph into a finite-dimensional Euclidean space.

Definition 6.22 (Admissible graph). Consider an Euclidean space (V,(−,−)). Anundirected weighted graph (N,E) with vertex set N ⊂V is admissible if

• its vertex set satisfiesN = v1, ...,vr ⊂V

with a linearly independent family (vi)i=1,...,r of unit vectors satisfying

(vi,v j)≤ 0 for 1≤ i 6= j ≤ r

• and its edge setE = (e,m) : e = vi,v j

satisfies

(vi,v j,m) ∈ E ⇐⇒ i 6= j and m = 4 · (vi,v j)2 ∈ 1,2,3.

Proof (of Theorem 6.21). The proof will classify all connected admissible graphs (N,E)arising from root systems Φ of rank r in an Euclidean space (V,(−,−)).

i) Any subgraph of an admissible graph obtained by removing a subset of verticesand their incident edges is admissible.

ii) An admissible graph has less edges than vertices, i.e. |E|< |N|:

Consider the element

v :=r

∑i=1

vi ∈V.

Then v 6= 0 because the family (vi)i=1,...,r is linearly independent. We obtain

0 < (v,v) =r

∑i=1

(vi,vi)+2 · ∑1≤i< j≤r

(vi,v j) = r+2 · ∑1≤i< j≤r

(vi,v j).

For an edge vi,v j with weight

m := 4 · (vi,v j)2 ∈ 1,2,3 and (vi,v j)≤ 0

follows2 · (vi,v j)≤−1,

170 6 Root systems

therefore0 < r−|E| i.e. |E|< |N|.

iii) An admissible graph has no cycles: A cycle is an admissible graph accordingto part i). But for a cycle the number of vertices equalizes the number of edges, incontradiction to part ii).

iv) For any vertex of an admissible graph the weighted sum of incident edges is atmost = 3:

Denote byInc(v) := (e,m) ∈ E : e incident with v

the set of weighted edges incident to a vertex v. We have to show

3≥ ∑(e,m)∈Inc(v)

m.

Denote by w1, ...,wk the set of vertices adjacent to v. Because an admissible graph

Fig. 6.3 Vertices adjacent to v

is cycle free due to part iii), we have

(wi,w j) = 0

for 1≤ i 6= j≤ k, i.e. the family (wi)i=1,...,k is orthonormal in (V,(−,−)). The family

(v,w1, ...,wk)

is linearly independent as a subfamily of the linearly independent family of all ver-tices. Choose a vector

w0 ∈ span < v,w1, ...,wk >

orthonormal to (wi)i=1,...,k and represent

v =k

∑i=0

(v,wi) ·wi.


The case (v,w0) = 0 is excluded, because v is linearly independent from (wi)i=1,...,k.Hence

(v,w0) 6= 0

and we obtain

1 = (v,v) =k

∑i=0

(v,wi)2

andk

∑i=1

(v,wi)2 < 1.

As a consequence

∑(e,m)∈Inc(v)

m =k

∑i=1

4 · (v,wi)2 < 4.

v) Blowing down a simple path, i.e. all its edges have weight = 1, results in a newadmissible graph (N′,E ′):

Fig. 6.4 Blow-down of a simple path

Denote by C = w1, ...,wn⊂N the vertices of the path. By assumption for i = 1, ...,n−1

4 · (wi,wi+1)2 = 1, i.e. 2 · (wi,wi+1) =−1.

The graph (N′,E ′), resulting from blowing down the original path, has

• vertex setN′ = (N \C)∪w0, w0 := ∑

w∈Cw,

• and edge setE ′ = E ′1∪E ′2

with

172 6 Root systems

E ′1 := (e,m) ∈ E : e = vi,v j,vi,v j ∈ N \C

E ′2 :=(e′,m) : e′ = v,w0,v ∈ (N \C) and exists (e,m) ∈ E with e = v,wi for i ∈ 1, ...,n

.

We show that the graph (N′,E ′) is admissible in the Euclidean subspace

V ′ := span < v1, ...,vr,w0 >⊂V

with the induced scalar product:

Linear independence of the vertex set N′ is obvious. Moreover, we compute

(w0,w0) =n

∑1≤i, j≤n

(wi,w j) =n

∑i=1

(wi,wi)+2 · ∑1≤i< j≤n

(wi,w j) =

n+2 ·n−1

∑i=1

(wi,wi+1) = n− (n−1) = 1.

In (N,E) any vertex w from N \C is adjacent to at most one vertex from the path,because an admissible graph is cycle free according to part iii). Hence

• either (w,w0) = 0• or exactly one index i = 1, ...,n exists with 0 6= (w,wi).

In either case holds4 · (w,w0)

2 ∈ 0,1,2,3.

vi) A connected admissible graph does not contain any subgraph from Figure 6.5:

a) Vertex with more than 3 adjacent vertices.

b) Two edges with weight = 2.

c) One edge with weight = 2 and one vertex with three adjacent vertices.

d) Two distinct vertices, both having three adjacent edges.

In any of the subgraphs b)-d) it would be possible to blow down a path to a vertexwith weighted sum of incident edges ≥ 4, which contradicts part v) and iv).

vii) Any conncected admissible graph belongs to one of the types from Figure 6.6:

a) All edges have weight = 1, no vertex has 3 incident edges.

b) A single edge has weight = 2, all other edges have weight = 1.

c) A single edge, it has weight 3.


Fig. 6.5 Non-admissible graphs and blow-down in case b)-d)

d) All edges have weight = 1, a single vertex has 3 incident edges, each withweight = 1.

The result follows from excluding the subgraphs from part vi).

viii) All connected admissible graphs from type vii,a) belong to series Ar. Obvious.

ix) All connected admissible graphs from type vii,b) belong to series Br or F4:

Consider the two vectors from V

u :=p

∑i=1

i ·ui and v :=q

∑i=1

i ·vi.

They are linearly independent. Using

2 · (ui,ui+1) =−1

we compute

(u,u) = ∑1≤i, j≤p

i · j · (ui,u j) =p

∑i=1

i2 · (ui,ui)+2 ·p−1

∑i=1

i(i+1) · (ui,ui+1) =

174 6 Root systems

Fig. 6.6 Admissible graphs


=p

∑i=1

i2 +2 ·p−1

∑i=1

(−1/2) · i(i+1) =p

∑i=1

i2−p−1

∑i=1

i2−p−1

∑i=1

i =

= p2− (1/2)p(p−1) = (p/2)(p+1).

Analogously(v,v) = (q/2)(q+1).

Because4 · (up,vq)

2 = 2

we obtain

(u,v)2 = (p ·up,q ·vq)2 = p2 ·q2 · (up,vq)

2 = (1/2)· p2 ·q2.

Employing the Cauchy-Schwarz inequality

(u,v)2 < (u,u) · (v,v)

with u and v linearly independent gives

(1/2)· p2q2 < (p/2)(p+1) · (q/2)(q+1).

Multiplying both sides by 2 implies

p2 ·q2 < (1/2)p(p+1)q(q+1) = (1/2)pq(p+1)(q+1)

pq < (1/2)(p+1)(q+1) = (1/2)pq+(1/2)(p+q+1)

pq < (p+q+1)

(p−1)(q−1)−2 < 0

and eventually(p−1)(q−1)< 2.

This restriction allows only the possibilities

(p,q) = (1,≥ 2),(p,q) = (≥ 2,1),(p,q) = (2,2),(p,q) = (1,1).

The first two give the same Coxeter graph. It has type Br,r ≥ 3, equal to type Cr.The third possibility gives F4. The fourth possibility is type B2.

x) The only connected admissible graph containing an edge with weight = 3 isthe graph G2: This result follows from part iv).

xi) All connected admissible graphs having type vii,d) belong to series Dr,r≥ 4,or series Er,r ∈ 6,7,8:

As in the proof of part ix) we define the vectors from V

176 6 Root systems

u :=r−1

∑i=1

i ·ui,v :=p−1

∑i=1

i · vi and w :=q−1

∑i=1

i ·wi.

Note r, p,q,≥ 2. The three vectors (u,v,w) are pairwise orthogonal and the four vec-tors (u,v,w,x) are linearly independent. Denote by θ1,θ2,θ3 the respective anglesbetween the vectors x and u,v,w. Similar to the proof of part iv) we obtain

1 >3

∑i=1

cos2θi.

Analogously to the proof of part x) we have

(u,u) = (r/2)(r−1),(v,v) = (p/2)(p−1),(w,w) = (q/2)(q−1).

Using in addition 4 · (x,ur−1)2 = 1 we obtain

cos2θ1 =

(x,u)2

‖x‖2 · ‖u‖2 =(r−1)2 · (x,ur−1))

2

‖u‖2 =(r−1)2 ·2 · (1/4)

r(r−1)= (1/2)(1−(1/r))

and analogously for cos2 θ2 and cos2 θ3. Hence

1 >3

∑i=1

cos2θi = (1/2)[(1− (1/r)+1− (1/p)+1− (1/q)]

or1/r+1/p+1/q > 1.

W.l.o.g we may assumeq≤ p≤ r

and q≥ 2 because q = 1 reproduces type An. Hence

1 < 1/r+1/p+1/q≤ 3/q

which implies3 > q≥ 2, i.e. q = 2.

We obtain1 < 1/r+1/p+1/2, i.e. 1/2 < 1/r+1/p,

and1/2 < 2/p,

hence2≤ p < 4.

In case p = 3 we have r < 6. In case p = 2 the parameter r may have any value ≥ 2.

Summing up: When r ≥ p≥ q then the only possibilities for (r, p,q) are


(5,3,2),(4,3,2),(3,3,2),(≥ 2,2,2).

These possibilities refer to the series E8,E7,E6 or to the series Dr,r ≥ 4, q.e.d.

Definition 6.23 (Irreducible root system). Consider a root system Φ of a vectorspace V and denote by (−,−) a scalar product on V invariant with respect to theWeyl group W of Φ .

1. The root system Φ is reducible iff it decomposes into two non-empty subsetswhich are orthogonal, i.e. iff a decomposition

Φ = Φ1∪Φ2,Φ1 6= /0,Φ2 6= /0,

exists with(Φ1,Φ2) = 0.

Otherwise Φ is irreducible.

2. Analogously defined are the terms reducible and irreducible for a base ∆ of Φ .

Proposition 6.24 (Irreducibility and connectedness of the Coxeter graph). Con-sider a root system Φ of a vector space V and a base ∆ of Φ .

i) Φ is irreducible if and only if ∆ is irreducible.

ii) Φ is irreducible if and only if its Coxeter graph is connected.

Proof. i) Suppose Φ reducible with decomposition

Φ = Φ1∪Φ2;Φ1 6= /0,Φ2 6= /0.

Define ∆i := Φi∩∆ , i = 1,2. Then

∆ = ∆1∪∆2

and (∆1,∆2) = 0.

Assume ∆1 = /0. Then ∆ = ∆2 ⊂Φ2 which implies

(Φ1,∆2)⊂ (Φ1,Φ2) = 0.

Because V = span ∆ = span ∆2 we even get

(Φ1,V ) = (Φ1,∆2) = 0.

Therefore

178 6 Root systems

Φ1 = 0,

which is excluded. As a consequence:

∆1 6= /0 and similarly ∆2 6= /0.

The decomposition∆ = ∆1∪∆2

proves the reducibility of ∆ .

For the opposite direction suppose ∆ reducible with decomposition

∆ = ∆1∪∆2.

Denote by W the Weyl group of Φ . Define

Φi := W (∆i), i = 1,2.

According to Lemma 6.16 any root β ∈Φ has the form β =w(α) for suitable w ∈Wand α ∈ ∆ . Therefore

Φ = Φ1∪Φ2.

The Weyl group is generated by the symmetries σα ,α ∈ ∆ . Explicit calculationshows:

• The orthogonality (∆1,∆2) = 0 implies that for two roots αi ∈ ∆i, i = 1,2, thecorresponding symmetries σα1 and σα2 commute.

• If α1 ∈ ∆1 then for a root α ∈ span ∆1 also σα1(α) ∈ span ∆1.• If α2 ∈ ∆2 then σα2(α) = α for any root α ∈ span ∆1.

As a consequence W (∆1)⊂ span ∆1 and similarly W (∆2)⊂ span ∆2. The orthog-onality (∆1,∆2) = 0 implies the orthogonality

(Φ1,Φ2) = 0, notably Φ1∩Φ2 = /0.

Because ∆ 6= /0 and id ∈ W also Φi 6= /0 i = 1,2. Therefore Φ is reducible withdecomposition

Φ = Φ1∪Φ2.

ii) The statement from part ii) is obvious, q.e.d.

The Dynkin diagram of a root system complements the Coxeter graph by theinformation about the length ratio of the elements from a base. This information canbe encoded by an orientation of the edges pointing from the long root to the shortroot in case two non-orthogonal roots have different length.


Definition 6.25 (Dynkin diagram of a root system). Consider a root system Φ andits Coxeter graph Coxeter(Φ) = (N,E). The Dynkin diagram of Φ is the directedweighted graph

Dynkin(Φ) := (N,ED)

with

• Vertex set: N = ∆ .

• Edge set: Edge (e,m) ∈ ED iff e = (α,β ) with

< α,β >< β ,α >6= 0 and‖α‖2

‖β‖2 =< α,β >

< β ,α >≥ 1.

In this case m :=< α,β >< β ,α > satisfies m ∈ 1,2,3.

Note: The Coxeter graph and the Dynkin diagram of a root system have the sameset of vertices. In the Dynkin diagram an oriented edge

e = (α,β )

is an ordered pair; the edge is considered oriented from the long root α to the shortroot β , ‖α‖ ≥ ‖β‖. If both roots have the same length, then also the edge withthe inverse orientation belongs to the Dynkin diagram. In this case the two ori-ented edges from the Dynkin diagram are equivalent to the corresponding singleundirected edge from the Coxeter graph. Therefore, often the figure of the Dynkindiagram suppresses the arrows of two opposite orientations.

Theorem 6.26 (Classification of connected Dynkin diagrams). Consider an irre-ducible root system Φ . Then its Dynkin diagram belongs to one the following types,see Figure 6.3:

• Series Ar,r ≥ 1

• Series Br,r ≥ 2

• Series Cr,r ≥ 3

• Series Dr,r ≥ 4

• Exceptional type G2

• Exceptional type F4

• Exceptional series Er,r ∈ 6,7,8

Notably, the two types Br and Cr are distinguished by the length ratio of their roots.

180 6 Root systems

Fig. 6.7 The Dynkin diagrams of irreducible root systems

Proof. The statement follows from the classification of Coxeter graphs according toTheorem 6.3 and the restriction of the length ratio of two roots from a base accordingto Lemma 6.7: A weighted edge (e,m) from the Coxeter graph links two roots withlength ratio 6= 1 if and only if m∈ 2,3. Therefore the Dynkin diagrams distinguishbetween the two series Br and Cr, q.e.d.

The Dynkin diagrams of an irreducible root system complements the Coxetergraph by the length ratio of the roots from a base. The Dynkin diagram contains the


full information of the Cartan matrix of the root system. Therefore from the Dynkindiagram of a root system Φ one can reconstruct a base ∆ and the Cartan matrix of Φ

by the algorithm from Proposition 6.27.

Proposition 6.27 (Characterisation of a root system by its Dynkin diagram).Consider a root system Φ of rank = r and its Dynkin diagram

Dynkin(Φ) = (N,ED).

A base ∆ of Φ and the Cartan matrix of Φ with respect to a numbering of theelements from ∆

Cartan(∆) = (< αi,α j >1≤i, j≤r)

can be recovered from the Dynkin diagram by the following algorithm:

1. Set ∆ := N.

2. For any vertex α ∈ N, i = 1, ...,r, set

< α,α >:= 2.

3. For a pair of vertices α,β ∈ N,α 6= β , not joined by an edge (e,m) ∈ ED set

< α,β >:= 0.

4. For any edge (e,m) ∈ ED oriented from α to β set

< α,β >:=−m,< β ,α >:=−1

Remember that an edge without orientation from the figure of a Dynkin diagramreplaces two oriented edges of the diagram.

Proof. The proof evaluates for each pair of roots the relation between the includedangle and the length ratio of the roots, see Lemma 6.7, q.e.d.

Chapter 7Classification

The objective of the present chapter is to classify complex semisimple Lie algebrasby their Dynkin diagram, more precisely the Dynkin diagram of its root system.The result is one of the highlights of Lie algebra theory. It completely encodes thestructure of these Lie algebras by a certain finite graph, a data structure from discretemathematics.

We consider a fixed pair (L,H) with a complex semisimple Lie algebra L and aCartan subalgebra H ⊂ L. We denote by Φ the root set corresponding to the Cartandecomposition of (L,H)

L = H⊕⊕α∈Φ

Lα .

From the Cartan criterion we know that a complex Lie algebra L is semisimple ifand only if its Killing form is non-degenerate. We even know from Proposition 5.15:Also the restriction of the Killing form to a Cartan subalgebra H

κH := κ|H×H→ C

is non-degenerate.

In this chapter L denotes a complex semisimple Lie algebra and H ⊂ L a Cartansubalgebra, if not stated otherwise.

7.1 The root system of a complex semisimple Lie algebra

Roots are linear functionals on H. Therefore we first translate properties of H intoproperties of its dual space H∗. This transfer is achieved by the non-degenerateKilling form κH :

Each linear functional λ : H → C is uniquely represented by an element tλ ∈ Hwith λ = κH(tλ ,−), i.e.

183

184 7 Classification

λ (h) = κH(tλ ,h) f or all h ∈ H.

The mapH ∼−→ H∗, tλ 7→ λ ,

is a linear isomorphism of complex vector spaces. In particular, for each root α ∈Φ ⊂ H∗

of L its inverse image under this isomorphism is denoted tα ∈ H, i.e.

α = κH(tα ,−).

This formula relates roots α ∈ Φ , being elements from the dual space H∗, to ele-ments tα ∈ H of the Cartan subalgebra.

Next we transfer the Killing form from a Cartan subalgebra H to a nondegeneratebilinear form on the dual space H∗

Definition 7.1 (Bilinear form on H∗). For the pair (L,H) the nondegenerate Killingform κH on H induces a nondegenerate symmetric bilinear form on the dual space H∗

(−,−) : H∗×H∗→ C,(λ ,µ) := κH(tλ , tµ).

Combining the definition of (λ ,µ) with the definition of tµ and tλ results in thefollowing formula

(λ ,µ) = κH(tλ , tµ) = λ (tµ) = µ(tλ ).

The objective of this section is to show that the root set Φ of (L,H) satisfies theaxioms for a root system of a real vector space V in the sense of Definition 6.2. Weconsider the real vector space

V := spanR Φ .

We will show that the root set Φ of (L,H) has the following properties:

• (R1) Finite and generating: |Φ |< ∞, 0 /∈Φ , dimR V = dimC H∗.

• (R2) Invariance under distinguished symmetries: Each α ∈Φ defines a symme-try

σα : V →V

with vector α ∈V and σα(Φ)⊂Φ .

• (R3) Cartan integers: The coefficients < β ,α >,α,β ∈Φ , from the representa-tion

σα(β ) = β−< β ,α > ·α

are integers, i.e. < β ,α >∈ Z.

7.1 The root system of a complex semisimple Lie algebra 185

• (R4) Reducedness: For each α ∈Φ holds Φ ∩R ·α = ±α.

To motivate the separate steps in deriving the root system of the Lie algebra L wefirst consider the example of sl(3,C).

Example 7.2 (The root system of sl(3,C)).

Set L := sl(3,C).

i) Cartan subalgebra: The subalgebra

H := span < h1 := E11−E22,h2 := E22−E33 >⊂ L

is a Cartan subalgebra with dim H = 2.

ii) Cartan decomposition: The Cartan decomposition of (L,H) is

L = H⊕⊕

α∈α1,α2,α3(Lα ⊕L−α).

All root spaces are 1-dimensional. The root set Φ = ±α1,±α2,±α3 satisfies:

• Lα1 = span < x1 := E12 >,L−α1 = span < y1 := E21 >

α1 : H→ C,α1(h1) = 2,α1(h2) =−1


α2 : H→ C,α2(h1) =−1,α2(h2) = 2


α3 : H→ C,α3(h1) = 1,α3(h2) = 1

Therefore α3 = α1 +α2.

iii) Killing form: To compute the Killing form κ and its restriction

κH : H×H→ C

one can use the formula

κ(z1,z2) = 2n · tr(z1 z2),z1,z2 ∈ sl(n,C),

with n = 3, see [28, Chapter 6, Ex. 7]. E.g.,

tr(h1 h2) = tr((E11−E22) (E22−E33)) = tr(−E22 E22) =−tr(E22) =−1.

Therefore

(κH(hi,h j)1≤i, j≤2) = 6 ·(

2 −1−1 2

).


iv) Bilinear form on H∗: With respect to the isomorphy

H ∼−→ H∗, tλ 7→ λ ,

we gettα1 = (1/6) ·h1, tα2 = (1/6) ·h2.

The family (α1,α2) is linearly independent and a basis of H∗. According toDefinition 7.1, on H∗ the induced bilinear form

(−,−) : H∗×H∗→ C

with respect to the basis (α1,α2) of V has the matrix

((αi,α j)1≤i, j≤2) = (αi(tα j)1≤i, j≤2) = (1/6) ·(

2 −1−1 2

).

In particular, all roots αi, i = 1,2,3, have the same lenght:

(αi,αi) = 2 · (1/6) = 1/3.

We define the real vector space

V := spanR < α1,α2 > .

ApparentlydimC V = dimC H∗ = 2.

The restriction of the bilinear form (−,−) to V is a real positive definite form, hencea scalar product. We now prove by explicit computation: The root set Φ of (L,H) isa root system of the real vector space V in the sense of Definition 6.2.

v) Distinguished symmetries: Consider the Euclidean space (V,(−,−)). Forfor i = 1,2,3 define the maps

σi : V →V,x 7→ x−6 · (x,αi) ·αi.

Here the factor 6 has been choosen in order to get

σi(αi) =−αi.

Moreover

σ1(α2) = α3,σ1(α3) = σ1(α1)+σ1(α2) =−α1 +α2 +α1 = α2

σ2(α1) = α3,σ2(α3) = α1


σ3(α1) = α1−6 · (α1,α3) ·α3 = α1− (α1 +α2) =−α2,

σ3(α2) = α2− (α1 +α2) =−α1.

For i = 1,2,3 we define the symmetries

σαi := σi and σ−αi :=−σαi ,

then σαi(α)⊂Φ for all α ∈Φ .

Each symmetry leaves the scalar product invariant, because for α ∈Φ ,x,y ∈V,

(σα(x),σα(y)) = (x−6(x,α) ·α,y−6(y,α) ·α) =

(x,y)−6(y,α)(x,α)−6(x,α)(α,y)+36(x,α)(y,α)(α,α) = (x,y)

using (α,α) = 1/3.

vi) Cartan integers: From the symmetries of part v) one reads off the Cartannumbers

< αi,αi >= 2, i = 1,2, and < α1,α2 >=< α2,α1 >=−1

which are integers indeed.

vii) Reducedness: Apparently for each α ∈Φ the only roots proportional to α

are ±α .

viii) Base: Apparently the set ∆ := α1,α2 is a base of Φ . The Cartan matrix withrespect to ∆ and the given numbering is

Cartan(∆) =

(2 −1−1 2

).

According to Lemma 6.7 both roots from ∆ have the same lenght and include theangle (2/3)π . Therefore Coxeter graph and Dynkin diagram of Φ contain the sameinformation. According to the classification from Theorem 6.26 the root systemof sl(3,C) has type A2. The scalar product (−,−) induced from the Killing form isinvariant with respect to the Weyl group W = span < σα1 ,σα2 >.

The following two propositions collect the main properties of the root set Φ of asemisimple complex Lie algebra L. They generalize the result from Proposition 5.4about the structure of sl(2,C).

Proposition 7.3 considers the complex linear structure of L and its canonicalsubalgebras sl(2,C) ⊂ L. While Proposition 7.4 considers the integrality proper-ties of the root set Φ with respect to the bilinear form induced by the Killing form,see Definition 7.1. These properties assure that Φ is a root system in the sense of


Definition 6.2. They allow to apply the classification of respectively Coxeter graphsfrom Theorem 6.21 and Dynkin diagrams from Theorem 6.26 to obtain the classifi-cation of complex semisimple Lie algebras in Proposition 7.7 - 7.10 and Section 7.3.

Recall from Corollary 5.13 that for any root α ∈Φ of L also the negative −α isa root of L.

Proposition 7.3 (Semisimple Lie algebras as sl(2,C)-modules). Consider a pair (L,H)with L a complex semisimple Lie algebra and H ⊂ L a Cartan subalgebra. Denoteby (−,−) the nondegenerate bilinear form on H∗ derived from the restricted Killingform κH on H according to Definition 7.1. The root set Φ of the Cartan decomposi-tion

L = H⊕⊕α∈Φ

Lα

has the following properties:

1. Generating: spanCΦ = H∗.

2. Duality of root spaces: For each α ∈ Φ the vector spaces Lα and L−α are dualwith respect to the Killing form, i.e. the bilinear map

Lα ×L−α → C,(x,y) 7→ κ(x,y),

is non-degenerate. The distinguished element tα ∈ H satisfies the equation

[x,y] = κ(x,y) · tα

for any x ∈ Lα ,y ∈ L−α . In particular,

[Lα ,L−α ] = C · tα .

3. Subalgebras sl(2,C): Consider an arbitrary but fixed α ∈Φ . The map

α : [Lα ,L−α ]−→ C, [x,y] 7→ κ(x,y),

is an isomorphism. If hα ∈ [Lα ,L−α ] denotes the uniquely determined elementwith α(hα) = 2 then

hα =2

(α,α)· tα .

For each non-zero element xα ∈ Lα a unique element yα ∈ L−α exists with

[xα ,yα ] = hα .

The remaining commutators are

[hα ,xα ] = 2 · xα , [hα ,yα ] =−2 · yα .


As a consequence, the subalgebra

Sα := spanC < hα ,xα ,yα >⊂ L

is isomorphic to sl(2,C) via the Lie algebra morphism

Sα

∼−→ sl(2,C)

hα 7→(

1 00 −1

),xα 7→

(0 10 0

),yα 7→

(0 01 0

).

The Lie algebra L is an Sα -module with respect to the module multiplication

Sα ×L→ L,(z,u) 7→ ad(z)(u) = [z,u]

which results from the adjoint representation of L.

Proof. ad 1) The Killing form defines an isomorphism H ∼−→ H∗. Assume on thecontrary that Φ does not span H∗. Then span Φ $ H∗ is a proper subspace. A non-zero linear functional

h ∈ (H∗)∗ ' H

exists which vanishes on this subspace, i.e.

h 6= 0 and α(h) = 0

for all α ∈Φ . We obtain [h,Lα ] = 0 for all α ∈Φ . In addition [h,H] = 0 because theCartan subalgebra H is Abelian, see Proposition 5.3. As a consequence [h,L] = 0,i.e.

h ∈ Z(L).

We obtain h = 0 because the center of the semisimple Lie algebra L is trivial. Thiscontradiction proves span Φ = H∗.

In order to prove the remaining issues we recall from Lemma 5.12 for two linearfunctionals λ ,µ ∈ H∗ with λ +µ 6= 0 the orthogonality

κ(Lλ ,Lµ) = 0.

As a consequence, for two roots α,β with β 6=−α , i.e. α +β 6= 0, holds

κ(Lα ,H) = 0 and κ(Lα ,Lβ ) = 0.

ad 2)

• If x ∈ Lα satisfies κ(x,L−α) = 0 then also κ(x,L) = 0, which implies x = 0.Hence the canonical map

Lα → (L−α)∗, x 7→ κ(x,−),


is injective. Interchanging the role of the two roots α and−α and using (Lα)∗∗ ' Lα

shows that the map is also surjective.

• We have[Lα ,L−α ]⊂ L0 = H

according to Lemma 5.12. The Killing form is associative according to Lemma 4.1.Hence for all h ∈ H, x ∈ Lα , y ∈ L−α

κ(h, [x,y]) = κ([h,x],y) = κ(α(h)x,y) = α(h)κ(x,y) = κ(tα ,h) ·κ(x,y) =

= κ(h, tα) ·κ(x,y) = κ(h,κ(x,y) · tα).

This equation is satisfied also by the restriced Killing form: For all h ∈ H, x ∈ Lα , y ∈ L−α

κH(h, [x,y]) = κH(h,κ(x,y) · tα).

Non-degenerateness of the restricted Killing form κH implies

[x,y] = κ(x,y) · tα .

The duality between Lα and L−α provides for each element xα ∈ Lα ,xα 6= 0, anelement yα ∈ L−α such that

κ(xα ,yα) 6= 0

proving in particular[Lα ,L−α ] = C · tα .

ad 3)

• For an arbitrary but fixed x ∈ Lα ,x 6= 0, one can choose an element y ∈ L−α

with κ(x,y) 6= 0 because Lα and L−α are dual according to part 2). If z := [x,y]then

z = κ(x,y) · tα 6= 0

according to the formula from part 3).

We claim α(z) 6= 0: Assume on the contrary α(z) = 0. Then define thesubalgebra

S :=< x,y,z >⊂ L.

Its Lie bracket satisfies

[z,x] = α(z)x = 0, [z,y] =−α(z)y = 0, [x,y] = z.

Apparently the Lie algebra S is nilpotent, in particular solvable. Semisimplicityof L implies that the adjoint representation

ad : L −→ gl(L)

embeds L and it subalgebra S into the matrix Lie algebra gl(L):


S' ad(S)⊂ gl(L).

According to Lie’s theorem, see Theorem 3.21, the solvable subalgebra ad(S)embeds into the subalgebra of upper triangular matrices. The element

ad(z) = ad[x,y],

being a commutator, is even a strict upper triangular matrix. Therefore ad(z) is anilpotent endomorphism, i.e. z ∈ L is ad-nilpotent. Because H is a Cartansubalgebra, the element z ∈ H is also ad-semisimple, hence z = 0. Thiscontradiction proves the claim.

Therefore the linear map

α : [Lα ,L−α ]→ C,h 7→ α(h),

between 1-dimensional vector spaces is an isomorphism. By definition

α(tα) = (α,α) and α(hα) = 2.

Hence

hα =2

(α,α)· tα .

• For an arbitrary element xα ∈ Lα , xα 6= 0, according to part 2) a uniqueelement yα ∈ L−α exists with

[xα ,yα ] = hα .

We obtain[hα ,xα ] = α(hα) · xα = 2 · xα

[h,yα ] =−α(hα) · yα =−2 · yα

[xα ,yα ] = hα .

Therefore the subalgebra Sα :=< xα ,yα ,hα > is isomorphic to sl(2,C), q.e.d.

Because L is a sl(2,C)-module with respect to each subalgebra Sα the resultsfrom section 5.2 about the structure of irreducible sl(2,C)-module apply. They im-ply a series of integrality and rationality properties of L, see Proposition 7.4. Fortwo roots α,β ∈Φ we will often employ the formula

β (hα) =2 ·β (tα)(α,α)

=2 ·κH(tβ , tα)

(α,α)=

2 · (β ,α)

(α,α).

It derives from the relation between hα and tα from Proposition 7.3 and from thedefining relation κH(tβ ,−) = β . The numbers β (hα) will turn out as the Cartanintegers < β ,α > of the root system. Notably they are integers.


Proposition 7.4 (Integrality and rationality properties of the root set). Considera pair (L,H) with L a complex semisimple Lie algebra and H ⊂ L a Cartan subal-gebra. The root set Φ from the Cartan decomposition

L = H⊕⊕α∈Φ

Lα

has the following properties:

1. Root spaces are 1-dimensional: dim Lα = 1 for each root α ∈Φ .

2. Integrality: For each pair α,β ∈Φ

β (hα) ∈ Z and β −β (hα) ·α ∈Φ .

3. Rationality and scalar product: Consider a basis B = (α1, ...,αr) of the complexvector space H∗ made up from roots αi ∈ Φ , i = 1, ...,r. Then any root β ∈ Φ isa rational combination of elements from B, i.e.

Φ ⊂VQ := spanQα1, ...,αr

anddimQ VQ = dimC H.

The bilinear form (−,−) restricts from H∗ to a rational form on the subspace VQ ⊂ H∗

and its restriction(−,−)Q : VQ×VQ→Q

is positive definite, i.e. a scalar product.

4. Symmetries: Extending scalars from Q to R defines the real vector space

V := R⊗QVQ = spanRΦ ⊂ H∗

and the scalar product (−,−)R on V . For each root α ∈Φ the map

σα : V →V,β 7→ σα(β ) := β −2 · (β ,α)R

(α,α)R·α,

is a symmetry of V with vector α and Cartan integers

< β ,α >= β (hα) ∈ Z,β ∈Φ .

It satisfies σα(Φ) ⊂ Φ . In addition: Each symmetry σα leaves invariant thescalar product (−,−)R.

5. Proportional roots: If α ∈Φ then the only roots proportional to α are ±α .


6. α-string through β : For each pair of non-proportional roots α,β ∈Φ let p,q ∈ Nbe the largest numbers such that

β − p ·α ∈Φ and β +q ·α ∈Φ .

Then all functionals from the string

β + k ·α ∈ H∗,−p≤ k ≤ q,

are roots.

Moreover, for two roots α ∈Φ ,β ∈Φ with α +β ∈Φ

[Lα ,Lβ ] = Lα+β .

Fig. 7.1 α-string through β

Proof. ad 1) According to Proposition 7.3, part 2) the root spaces Lα and L−α aredual with respect to the Killing form κ .

In order to show dim Lα = 1 we assume on the contrary

dim Lα = dim L−α > 1.

Consider an element xα ∈ Lα ,x 6= 0, and the correponding subalgebra Sα fromProposition 7.3, part 3). An element hα ∈ Sα exists with [hα ,xα ] = 2 · xα , i.e.

α(hα) = 2.

The linear functional κ(xα ,−) on L−α is non-zero. Because dim L−α > 1 the func-tional has a non-trivial kernel, i.e. an element y∈L−α ,y 6= 0, exists with κ(xα ,y) = 0.The latter formula implies [xα ,y] = 0, see Proposition 7.3, part 2). As a consequence y ∈ Lis a primitive element of an irreducible Sα -submodule of L with weight

(−α)(hα) =−2 < 0.

The latter property contradicts the fact that all irreducible sl(2,C)-modules have anon-negative highest weight, see Theorem 5.8.


ad 2) Consider an element 0 6= y ∈ Lβ . Considered as an element of the rootspace Lβ it satisfies

[hα ,y] = β (hα) · y.

When considered as an element of the Sα -module L the element y∈L has weight β (hα).Theorem 5.8 implies

β (hα) ∈ Z

We set p := β (hα) and

z := ypα .y ∈ L if p≥ 0 and z := x−p

α .y ∈ L if p≤ 0.

Theorem 5.8 implies z 6= 0. Because

z ∈ Lβ−p·α

we get β −β (hα) ·α ∈Φ .

ad 3)

• Each root β ∈Φ has a unique representation

β =r

∑i=1

ci ·αi

with complex coefficients ci ∈ C, i = 1, ...,r. We claim that the coefficients areeven rational, i.e. ci ∈Q for i = 1, ...,r.

Multiplying the representation of β successively for j = 1, ...,r by

2(α j,α j)

and applying the bilinear form (−,α j) to the resulting equation gives a linearsystem of equations

b = A · c

for the vector of indeterminates c := (c1, ...,cr)>. The left-hand side is the vector

b =

(2 · (β ,α j

(α j,α j)

)>and the coefficient matrix is

A =

(2 · (αi,α j)

(α j,α j)

), row index j, column index i.

The system is defined over the ring Z because for two roots γ,δ ∈Φ


2 ·(γ,δ )

(δ ,δ )= γ(hδ ) ∈ Z

according to part 2). The coefficient matrix

A ∈M(r× r,Z)

is invertible as an element from GL(r,Q): It originates by multiplying for j = 1, ...,rthe row with index j of the matrix

(αi,α j)1≤i, j≤r ∈ GL(r,C)

by the non-zero scalar 2/(α j,α j). And the latter matrix defines the bilinearform (−,−) on H∗, which is non-degenerate because it corresponds to theKilling form κH(−,−) on H. As a consequence, the solution of the linear systemof equations is unique and is already defined over the base field Q, i.e.

c j ∈Q for all j = 1, ...,r.

• We claim that the bilinear form (−,−)Q is rational, i.e. it is defined over thefield Q, and positive definite. For any two roots α,β ∈Φ we have to show

(α,β ) ∈Q and (α,α)> 0.

For two linear functionals λ ,µ ∈ H∗ we compute

(λ ,µ) = κH(tλ , tµ) = tr(ad tλ ad tµ).

In order to evaluate the trace we employ the defining property of a root space: Ifz ∈ Lγ then

(ad tµ)(z) = γ(tµ) · z

and(ad tλ ad tµ)(z) = γ(tλ ) · γ(tµ) · z.

Using the Cartan decomposition

L = H⊕⊕γ∈Φ

Lγ

and observing [H,H] = 0 we obtain

(λ ,µ) = tr(ad tλ ad tµ) = ∑γ∈Φ

γ(tλ ) · γ(tµ).

We apply this formula to the computation of (α,α) using

2 · (γ,α)

(α,α)= 2 ·

γ(tα)(α,α)

= γ(hα) =:< γ,α >∈ Z


according to part 2). As a consequence

4 · γ(tα)2 = (α,α)2·< γ,α >2 .

We obtain

(α,α) = ∑γ∈Φ

γ(tα)2 = (1/4) · (α,α)2 · ∑γ∈Φ

< γ,α >2

and because of (α,α) 6= 0

1 = (1/4) · (α,α) · ∑γ∈Φ

< γ,α >2 .

In particular

(α,α) = 4 · ( ∑γ∈Φ

< γ,α >2)−1 ∈Q and (α,α)> 0

and(β ,α) = (1/2)(α,α)·< β ,α >∈Q.

ad 4) Due to the characterization α(hα) = 2 from Proposition 7.3 the map σα is asymmetry of V with vector α . Due to part 2) its Cartan numbers are integers

< β ,α >=2 · (β ,α)

(α,α)= β (hα) ∈ Z.

The inclusion σα(Φ)⊂Φ has been proven in part 2). In order to prove theinvariance of the scalar product with respect to the Weyl group it suffices toconsider three roots α,β ,γ ∈Φ :

(σα(β ),σα(γ)) = (β −β (hα)α,γ− γ(hα) ·α) =

= (β ,γ)−β (hα)(α,γ)− γ(hα)(β ,α)+β (hα)γ(hα)(α,α)

After using

(α,γ) = (1/2)(α,α)γ(hα) and (α,β ) = (1/2)(α,α)β (hα)

we obtain(σα(β ),σα(γ)) = (β ,γ).

ad 5) Assume on the contrary the existence of a pair of proportional roots differentfrom ±α . W.l.o.g. we may assume α ∈Φ and β = t ·α ∈Φ ,0 < t < 1, t ∈ R.From β (hα) ∈ Z and

β (hα) ·α = σα(β )−β = t · (σα(α)−α) =−2t ·α

follows


2t ∈ Z.

Hence the two roots from any pair of proportional roots belong to a set

±α,±(1/2)α,α ∈Φ .

We may now assume α ∈Φ and check whether the linear functional 2α ∈ H∗

belongs to Φ : Consider an element z ∈ L2α . Then

[hα ,z] = 2α(hα) · z = 4 · z.

Because 3α /∈Φ we have xα .z = 0.

With respect to the sl(2,C)-module structure of L induced from the action of

Sα =< xα ,yα ,hα >,

see Proposition 7.3, part 3), we have

• hα .z = 2 ·2 · z, because z ∈ L2α .

• xα .z ∈ L3α . But 3α /∈Φ , hence xα .z = 0.

• yα .(xα .z) ∈ Lα , hence yα .(xα .z) ∈ Cxα .

As a consequence

4z = hα .z = [xα ,yα ].z = xα .(yα .z)− yα .(xα .z) = yα(xα .z) ∈ C · xα ∈ L2α ∩Lα

which implies z = 0. The conclusion L2α = 0 implies 2α /∈Φ .

ad 6) Consider the Sα -module

M :=q⊕

k=−p

Lβ+kα .

Non-zero elements from Lβ+kα have - with respect to Sα - the weight

(β + kα)(hα) = β (hα)+2k.

The functional β + kα is a root of L if and only if Lβ+kα 6= 0. Then Lβ+kα

is 1-dimensional according to part 1). Proposition 5.6 implies that M is anirreducible Sα -module of dimension

m+1 = p+q+1.

In particular, for every −p≤ k ≤ q the number β (hα)+2k is a weight of M andevery functional β + kα is a root of L.


In addition, Proposition 5.6 implies that the shift of weight spaces of M

xα : Lβ → Lβ+α ,v 7→ xα .v,

is an isomorphism, q.e.d.

Theorem 7.5 (Root system of a complex simple Lie algebra). Consider a pair (L,H)with L a complex semisimple Lie algebra and H ⊂ L a Cartan subalgebra.

The root set Φ from the Cartan decomposition

L = H⊕⊕α∈Φ

Lα

is a root system in the sense of Definition 6.2 of the real vector space

V := spanRΦ ⊂ (H∗)R.

It is named the root system of the Lie algebra L. The root system attaches toeach α ∈Φ the symmetry

σα : V →V,β 7→ β −β (hα) ·α

with Cartan integers< β ,α >= β (hα) ∈ Z.

The bilinear form (−,−) on V , defined as

(α,β ) := κ(tα , tβ ),α,β ∈Φ ,

is an invariant scalar product with respect to the Weyl group of Φ .

Proof. The proof collects the results from Proposition 7.4, q.e.d.

7.2 Root systems of the A,B,C,D-series

We will show in the present section that the root systems of type Ar,Br,Cr,Dr arethe root systems of the complex Lie algebras belonging to the classical groups ofthe correponding types, see Proposition 2.11. We show the simpleness of these Liealgebras as a consequence of their Cartan decomposition. We follow [22, Chap. 7.7]and [25, Chap. X, §3].

The Lie algebras of the classical groups are subalgebras of the Lie algebra gl(n,C).We introduce the following notation for the elements of the canonical basis of thevector space M(n×n,C):

7.2 Root systems of the A,B,C,D-series 199

Ei, j ∈M(n×n,C)

is the matrix with entry = 1 at place (i, j) and entry = 0 for all other places. Ourmatrix computations are based on the formulas

Ei, j ·Ek,l = δ j,k ·Ei,l ,1≤ i, j,k, l ≤ n.

The family(Ei,i)1≤i≤n

is a basis of the subspace of diagonal matrices d(n,C). Denote the elements of thedual base by

εi := Ei,i∗ ∈ d(n,C)∗, i = 1, ...,n.

First we show, that the classical Lie algebras are semisimple.

Proposition 7.6 (Semisimpleness of the Lie algebras of type Ar,Br,Cr,Dr). Thecomplex Lie algebras of type Ar,Br,Cr,Dr are semisimple.

Note. We already know from Proposition 4.7 that the Lie algebras of type Ar aresemi-simple. The following proof is a refined form of the proof from Proposition 4.7and holds for all classical Lie algebras of the above types.

Proof. Denote by L ∈ M(n× n,C) a complex Lie algebra from one of the abovetypes. Due to the definition of the Lie algebra L all matrices of L are traceless, i.e.

L⊂ sl(n,C).

We have to show rad(L) = 0. Due to its definition the Lie algebra L contains withany matrix also its transpose:

A ∈ L =⇒ A> ∈ L.

Because the radical is solvable we may assume according to Theorem 3.21

rad(L)⊂ t(n,C)∩L.

Becauserad(L)> := A> : A ∈ rad(L)

is also solvable and rad(L) is maximal solvable, we have

rad(L)> ⊂ rad(L),

and after transposition also

rad(L)⊂ rad(L)>.

As a consequence


rad(L) = rad(L)>

is also contained in the set of lower triangular matrices. Therefore rad(L) is con-tained in the diagonal matrices

rad(L)⊂ d(n,C)∩L.

Because rad(L)⊂ L is an ideal, each

A =n

∑i=1

ai ·Ei,i ∈ rad(L), ai ∈ C for 1≤ i≤ n,

satisfies for all B ∈ L[A,B] ∈ rad(L)⊂ d(n,C)∩L.

For each arbitrary, but fixed pair of indices

( j0,k0), 1≤ j0 6= k0 ≤ n,

we choose an element

B =n

∑j,k=1

b j,k ·E j,k ∈ L, b j,k ∈ C for 1≤ j,k ≤ n,

with b j0,k0 6= 0. Such choice is possible, cf. the normal forms of the matrices from Lin [28, Ch. I.2]. Then

[A,B] =n

∑i, j,k=1

ai ·b j,k · [Ei,i,E j,k] =n

∑j,k=1

(a j−ak)b j,k ·E j,k ∈ d(n,C)∩L.

As a consequence(a j0 −ak0)b j0,k0 = 0

and a j0 = ak0 . Varying now the indices j0 6= k0 we obtain

a1 = ...= an

i.e. for a suitable a ∈ CA = a ·1.

Because tr A = 0 we get A = 0, which finishes the proof, q.e.d.

Proposition 7.7 (Typ Ar). The Lie algebra L := sl(r+1,C),r≥ 1, has the followingcharacteristics:

1. dim L = (r+1)2−1.


2. The subalgebraH := d(r+1,C)∩L

is a Cartan subalgebra with dim H = r.

3. The family (hi)i=1,...,r with

hi := Ei,i−Ei+1,i+1

is a basis of H.

4. Define the functionals

εi := (εi|H) ∈ H∗, i = 1, ...,r+1.

Then the root system Φ of L has the elements

εi− ε j,1≤ i 6= j ≤ r+1.

The corresponding root spaces are 1-dimensional, generated by the elements

Ei j,1≤ i 6= j ≤ r+1.

A base of Φ is the set ∆ := αi : 1≤ i≤ r with

αi := εi− εi+1.

The positive roots are the elements of Φ+ = εi − ε j : i < j. They have therepresentation

εi− ε j =j−1

∑k=i

αk ∈Φ+.

5. For each positive root α := εi− ε j ∈Φ+ the subalgebra

Sα ' sl(2,C)

is generated by the three elements

hα := Ei,i−E j, j,xα := Ei, j,yα := E j,i.

6. The Cartan matrix of Φ referring to the base ∆ is

Cartan(∆) =

2 −1 0 . . . 0−1 2 −1 . . . 0

. . .. . .

. . .0 . . . −1 2 −10 . . . 0 −1 2

∈M((r+1)× (r+1),Z).


All roots α ∈ ∆ have the same length. The only pairs of roots from ∆ , which arenot orthogonal, are

(αi,αi+1), i = 1, ...,r−1.

Each pair includes the angle2π

3. In particular the Dynkin diagram of the root

system Φ has type Ar from Theorem 6.26.

Proof. 4) For i 6= j elements h ∈ H act on Ei, j according to

[h,Ei, j] = hEi, j−Ei, j h = εi(h) ·Ei, j− ε j(h) ·Ei, j = (εi(h)− ε j(h)) ·Ei, j

5) According to part 4) the commutators are

[hα ,xα ] = [hα ,Ei, j] = (εi(hα)− ε j(hα))Ei, j = 2 ·Ei, j = 2 · xα

[hα ,yα ] = [hα ,E j,i] = (ε j(hα)− εi(hα))E j,i =−2 ·Ei, j =−2 · yα

[xα ,yα ] = [Ei, j,E j,i] = Ei,i−E j, j = hα .

6) The Cartan matrix has the entries β (hα),α,β ∈ ∆ . We have to consider

• βk = εk− εk+1,1≤ k ≤ r and• α j = ε j− ε j+1,1≤ j ≤ r, with corresponding elements h j = E j, j−E j+1, j+1.

βk(hα j) = (εk− εk+1)(E j, j−E j+1, j+1) = (εk− εk+1)(E j, j−E j+1, j+1) =

= δk j−δk, j+1−δk+1, j +δk+1, j+1 = 2δk j−δk, j+1−δk+1, j =

=

2, if k = j−1, if |k− j|= 1

0, if |k− j| ≥ 2

If α 6= β and <α,β >,< β ,α >6= 0 then the symmetry of the Cartan matrix implies

1 =< β ,α,>

< α,β >=‖β‖2

‖α‖2

Hence all simple roots have the same length. The angles between two simple rootscan be read off from the Cartan matrix. Hence the root system Φ has the Dynkindiagram Ar from Theorem 6.26, q.e.d.

In dealing with the Lie algebra sp(2r,C) we note that X ∈ sp(2r,C) iff

σ ·X> ·σ = X

with

σ :=(

0 1

−1 0

)and σ

−1 =−σ .

This condition is equivalent to


X =

(A BC −A>

)with symmetric matrices B = B> and C =C>. The following proposition will referto this type of decomposition.

Proposition 7.8 (Typ Cr). The Lie algebra L := sp(2r,C),r ≥ 3, has the followingcharacteristics:

1. dim L = r(2r+1).

2. The subalgebra

H :=(

D 00 −D

)∈ L : D ∈ d(r,C)


3. The family(hi := Ei,i−Er+i,r+i)1≤i≤r

is a basis of H.

4. Define the functionals

εi := εi|H ∈ H∗, i = 1, ...,2r.

Then the elements

Ei, j−Er+ j,r+i,1≤ i 6= j ≤ r, (Matrix o f type A),

Ei,r+ j +E j,r+i,1≤ i, j ≤ r, (Matrix o f type B),

Er+i, j +Er+ j,i,1≤ i, j ≤ r, (Matrix o f type C),

generate the 1-dimensional root spaces belonging to the respective root

α =

εi− ε j type Aεi + ε j type B−εi− ε j type C

The root system Φ is

Φ = εi−ε j : 1≤ i 6= j≤ r∪±(εi +ε j) : 1≤ i < j≤ r∪±2 ·εi : 1≤ i≤ r

A base of Φ is the set∆ := α1, ...,αr

with - note the different form of the last root αr -


• α j := ε j− ε j+1,1≤ j ≤ r−1,• αr = 2 · εr

The set of positive roots is

Φ+ = εi± ε j : 1≤ i < j ≤ r∪2 · ε j : 1≤ i≤ r.

5. For each positive root α ∈Φ+ the subalgebra

Sα =< hα ,xα ,yα >' sl(2,C)

has the generators:

• Typ A: If α := εi− ε j ∈Φ+,1≤ i < j ≤ r, then

hα := Ei,i−E j, j,xα := Ei, j−Er+ j,r+i,yα := E j,i−Er+i,r+ j.

• Typ B: If α := εi + ε j ∈Φ+,1≤ i < j ≤ r, then

hα := Ei,i +E j, j,xα := Ei,r+ j +E j,r+i,yα := Er+i, j +Er+ j,i.

• Typ A: If α := 2εi ∈Φ ,1≤ i≤ r, then

hα := Ei,i−Er+i,r+i,xα := Ei,r+i,yα := Er+i,i.

6. The Cartan matrix of the root system Φ referring to the basis ∆ is

Cartan(∆) =

2 −1 0 . . . 0−1 2 −1 . . . 0

. . .. . .

. . .0 . . . −1 2 −1 00 . . . 0 −1 2 −10 . . . 0 −2 2

∈M(r× r,Z),

entry <αi,α j > at position (row,column)= ( j, i). Note the distinguished entry −2in the last row: The Cartan matrix is not symmetric.All roots α j ∈ ∆ ,1 ≤ j ≤ r− 1 have equal length, they are the short roots. Theroot αr is the long root:

√2 =‖αr‖‖α j‖

,1≤ j ≤ r−1.

The only pairs of simple roots with are not orthogonal are

(αi,αi+1), i = 1, ...,r−1.


For i = 1, ...,r− 2 these pairs include the angle2π

3, while the pair (αr−1,αr)

inludes the angle3π

4. In particular the Dynkin diagram of the root system Φ has

type Cr according to Theorem 6.26.

Proof. 1) Using the representation

X =

(A BC −A>

)∈ sp(2r,C)

with symmetric matrices B = B> and C = C> we obtain from the number of freeparameters for A,B,C

dim L = r2 +2 · (r2− r

2+ r) = 2r2 + r = r(2r+1).

4) Note εi(h) =−εr+i(h) for all h ∈ H. For h ∈ H the commutators are:

[h,Ei, j−Er+ j,r+i] = h · (Ei, j−Er+ j,r+i)− (Ei, j−Er+ j,r+i) ·h =

= εi(h)Ei, j− εr+ j(h)Er+ j,r+i− ε j(h)Ei, j + εr+i(h)Er+ j,r+i =

= (εi(h)− ε j(h))Ei, j− εr+ j− εr+i)Er+ j,r+i =

= (εi(h)− ε j(h))Ei, j + ε j(h)− εi(h))Er+ j,r+i =

(εi(h)− ε j(h))(Ei, j−Er+ j,r+i)

[h,Ei,r+ j +E j,r+i] = εi(h)Ei,r+ j + ε j(h)E j,r+i− εr+ j(h)Ei,r+ j− εr+i(h)E j,r+i =

(εi(h)− εr+ j(h))Ei,r+ j +(ε j(h)− εr+i(h))E j,r+i =

= (εi(h)+ ε j(h))Ei,r+ j +(ε j(h)+ εi(h))E j,r+i =

(εi(h)+ ε j(h))(Ei,r+ j +E j,r+i)

[h,Er+i, j +Er+ j,i] = εr+i(h)Er+i, j + εr+ j(h)Er+ j,i− ε j(h)Er+i, j− εi(h)Er+ j,i =

= (εr+i(h)− ε j(h))Er+i, j +(εr+ j(h)− εi(h))Er+ j,i =

= (−εi(h)− ε j(h))Er+i, j +(ε j− εi(h))Er+ j,i =

= (−εi(h)− ε j(h))(Er+i, j +Er+ j,i)

The positive roots have the base representation

εi− ε j =j−1

∑k=i

αk,1≤ i < j ≤ r.


2 · εi = αr +r−1

∑k=i

2 ·αk,1≤ i≤ r−1.

Part 6) The Cartan matrix has the entries β (hα),α,β ∈ ∆ . We have to consider theroots

• βk = εk− εk+1,1≤ k ≤ r−1, and• βr = 2εr

and the roots

• α j = ε j− ε j+1,1≤ j ≤ r−1, with elements hα j = h j−h j+1 and• αr = 2εr with element hαr = hr.

Accordingly, we calculate the cases:

• For 1≤ j,k ≤ r−1:

βk(hα j) = (εk− εk+1)(E j, j−E j+1, j+1) = δk, j−δk, j+1−δk+1, j +δk+1, j+1 =

= 2 ·δk, j−δk, j+1−δk, j−1

• For 1≤ j ≤ r−1,k = r:

βr(hα j) = 2 · εr(E j, j−E j+1, j+1) = 2(δr, j−δr, j+1) =−2δr, j+1

• For 1≤ k ≤ r−1, j = r:

βk(hαr) = (εk− εk+1)(Er,r) = δk,r−δk+1,r =−δk+1,r

• For j = k = r:βr(hαr) = 2 · εr(Er,r) = 2.

If α 6= β and < α,β >,< β ,α >6= 0 then

β (hα)

α(hβ )=‖α‖2

‖β‖2

Hence

2 =−2−1

=αr(hαr−1)

αr−1(hαr)=‖αr−1‖2

‖αr‖2 ,q.e.d.

In dealing with the Lie algebra so(2r,C) it is useful to consider a matrix M ∈ so(2r,C)as a scheme having 2×2-matrices as entries. Hence we introduce the non-commutativering

R := M(2×2,C)

and consider the matrices

M = ((a jk)1≤ j,k≤r) ∈M(r× r,R)'M(2r×2r,C)


with entriesa jk ∈ R,1≤ j,k ≤ r.

For each 1≤ j,k ≤ r we introduce the matrix

E j,k ∈M(r× r,R)

with onyl one nonzero entry, namely 1 ∈ R at place ( j,k). We distinguish the fol-lowing elements from R

h :=(

0 −ii 0

), s :=

12

(i −1−1 −i

), t :=

12

(i 11 −i

), u :=

12

(i 1−1 i

), v :=

12

(i −11 i

)satisfying

• s = s>,h · s = s,s ·h =−s

• t = t>,h · t =−t, t ·h = t

• u> = v,h ·u = u ·h = u

• v> = u,h · v = v ·h =−v

• [s, t] =−h

• u2− v2 =−h

Proposition 7.9 (Typ Dr). The Lie algebra L := so(2r,C),r ≥ 4, has the followingcharacteristics:

1. dim L = r(2r−1).

2. The subalgebra

H := spanC < h ·E j, j : 1≤ j ≤ r > ⊂ d(r,R)∩L


3. For any pair 1≤ j < k ≤ r each of the four elements

s · (E j,k−Ek, j), t · (E j,k−Ek, j),u ·E j,k− v ·Ek, j,v ·E j,k−u ·Ek, j

generates the 1-dimensional root space belonging to the respective root

α =

ε j + εk

−ε j− εk

ε j− εk

−ε j + εk


Here ε j := (h ·E j, j)∗ ⊂ H∗,1≤ j ≤ r are the dual functionals.

4. The root system Φ of L has the elements

−εk− εn,εk + εn,−εk + εn,εk− εn,1≤ k < n≤ r,

for short

Φ = ±εk± εn : 1≤ k < n≤ r (each combination o f signs).

A base of Φ is the set∆ := α1, ...,αr

with - note the different form of the last root αr -

• α j := ε j− ε j+1,1≤ j ≤ r−1,• αr = εr−1 + εr

The set of positive roots is Φ+ = εk± εn : 1≤ k < n≤ r.

5. For each positive root α := ε j + εk ∈Φ+,1≤ j < k ≤ r, the subalgebra

Sα ' sl(2,C)


hα := h · (E j, j +Ek,k),xα := s · (E j,k−Ek, j),yα := t · (E j,k−Ek, j).

For each positive root α := ε j− εk ∈Φ+,1≤ j < k ≤ r, the subalgebra

Sα ' sl(2,C)


hα := h · (E j, j−Ek,k),xα := u ·E j,k− v ·Ek, j,yα := v ·E j,k−u ·Ek, j.

6. The Cartan matrix of the root system Φ referring to the basis ∆ is

Cartan(∆) =

2 −1 0 . . . 0−1 2 −1 . . . 0

. . .. . .

. . .0 . . . −1 2 −1 −10 . . . 0 −1 2 00 −1 0 2

∈M(r× r,Z).

Note the distinguished entries in the last row and in the last column.All roots α ∈ ∆ have the same length. The only pairs of simple roots with are notorthogonal are


(αi,αi+1), i = 1, ...,r−2, and (αr−2,αr).

These pairs include the angle2π

3. In particular the Dynkin diagram of the root

system Φ has type Dr from Theorem 6.26.

Proof. 3) It suffices to consider only pairs of indices (k, l) with k < l because thegenerators belonging to (k, l) and (l,k) differ only by sign.

The general element of H has the form

z =r

∑ν=1

aν ·h ·Eν ,ν ,aν ∈ C.

H acts on s · (E j,k−Ek, j) according to

[z,s ·(E j,k−Ek, j)] = ε j(z) ·hs ·E j,k−εk(z) ·hs ·Ek, j−εk(z) ·sh ·E j,k+ε j(z) ·sh ·Ek, j =

ε j(z) · s ·E j,k− εk(z) · s ·Ek, j + εk(z) · s ·E j,k− ε j(z) · s ·Ek, j =

= (ε j(z)+ εk(z)) · s · (E j,k−Ek, j)

H acts on t · (E j,k−Ek, j) according to

[z, t ·(E j,k−Ek, j)] = ε j(z) ·ht ·E j,k−εk(z) ·ht ·Ek, j−εk(z) ·th ·E j,k+ε j(z) ·th ·Ek, j =

−ε j(z) · t ·E j,k + εk(z) · t ·Ek, j− εk(z) · t ·E j,k + ε j(z) · t ·Ek, j =

= (−ε j(z)− εk(z)) · t · (E j,k−Ek, j)

H acts on u ·E j,k− v ·Ek, j according to

[z,u ·E j,k−v ·Ek, j] = ε j(z)·hu·E j,k−εk(z)·hv·Ek, j−εk(z)·uh ·E j,k+ε j(z)·vh·Ek, j =

ε j(z) ·u ·E j,k + εk(z) · v ·Ek, j− εk(z) ·u ·E j,k− ε j(z) · v ·Ek, j =

= (ε j(z)− εk(z)) · (u ·E j,k− v ·Ek, j)

H acts on v ·E j,k−u ·Ek, j according to

[z,v ·E j,k−u·Ek, j] = ε j(z)·hv ·E j,k−εk(z)·hu ·Ek, j−εk(z)·vh·E j,k+ε j(z)·uh ·Ek, j =

−ε j(z) ·u ·E j,k− εk(z) ·u ·Ek, j + εk(z) · v ·E j,k + ε j(z) ·u ·Ek, j =

= (−ε j(z)− εk(z)) · (v ·E j,k−u ·Ek, j).

The positive roots have the base representation

εi− ε j =j−1

∑k=i

αk,1≤ i < j ≤ r,


εi + ε j =r−2

∑k=i

αk +r

∑k= j

αk,1≤ i < j ≤ r.

5) For α := ε j + εk ∈Φ+,1≤ j < k ≤ r, the commutators are

[hα ,xα ] = [h·(E j, j+Ek,k),s·(E j,k−Ek, j)]= hs·(E j, j+Ek,k)(E j,k−Ek, j)−sh·(E j,k−Ek, j)(E j, j+Ek,k)=

= s · (E j,k−Ek, j)+ s · (E j,k−Ek, j) = 2 · xα

[hα ,yα ] = [h ·(E j, j+Ek,k), t ·(E j,k−Ek, j)]= −t ·(E j,k−Ek, j)−t ·(E j,k−Ek, j)=−2 ·yα

[xα ,yα ] = [s ·(E j,k−Ek, j), t(E j,k−Ek, j)]= [s, t]·(E j,k−Ek, j)2 =(−h)·(−E j, j−Ek,k)= hα

For α := ε j− εk ∈Φ+,1≤ j < k ≤ r, the commutators are

[hα ,xα ] = [h ·(E j, j−Ek,k),u·E j,k−v·Ek, j)= hu ·E j,k+hv·Ek, j+uh ·E j,k+vh·Ek, j =

u ·E j,k− v ·Ek, j +u ·E j,k− v ·Ek, j = 2u ·E j,k−2v ·Ek, j = 2 · xα

[hα ,yα ] = [h ·(E j, j−Ek,k),v ·E j,k−u ·Ek, j] = hv·E j,k+hu ·Ek, j+vh·E j,k+uh ·Ek, j =

−v ·E j,k +u ·Ek, j− v ·E j,k +u ·Ek, j = 2u ·Ek, j−2v ·E j,k =−2 · yα

[xα ,yα ] = [u ·E j,k−v ·Ek, j,v ·E j,k−u ·Ek, j] = −u2E j, j−v2Ek,k +v2E j, j +u2Ek,k =

= (v2−u2) ·E j, j +(u2− v2) ·Ek,k = h · (E j, j−Ek,k) = hα

6) The Cartan matrix has the entries β (hα),α,β ∈ ∆ . We have to consider

• α j = ε j− ε j+1,1≤ j ≤ r−1 with elements hα j = h · (E j, j−E j+1, j+1) and• αr = εr−1 + εr with element hαr = h · (Er−1,r−1 +Er,r)

and

• βk = εk− εk+1,1≤ k ≤ r−1 and• βr = εr−1 + εr.

If 1≤ j,k ≤ r−1 then

(εk−εk+1)(h·(E j, j−E j+1, j+1))= δk j−δk, j+1−δk+1, j+δk+1, j+1 = 2δk j−δk, j+1−δk+1, j =

=

2, if i = j−1, if |i− j|= 1

0, if |i− j| ≥ 2

If k = r and 1≤ j ≤ r−1 then

(εr−1 + εr)(h · (E j, j−E j+1, j+1)) = δr−1, j−δr−1, j+1 +δr, j−δr, j+1 =−δr, j+2


=

−1, if j = r−20, if j 6= r−2

If 1≤ k ≤ r−1 and j = r then

(εk− εk+1)(h · (Er−1,r−1 +Er,r)) = δk,r−1 +δk,r−δk+1,r−1−δk+1,r =−δk,r−2

=

−1, if k = r−2

0, if k 6= r−2

If k = r = 2 then

(εr−1 + εr)(h · (Er−1,r−1−Er,r)) = δr−1,r−1 +δr,r = 2.

If α 6= β and < α,β >,< β ,α >6= 0 then

< α,β >

< β ,α >=‖β‖2

‖α‖2, q.e.d.

In order to investigate the Lie algebra L := so(2r+ 1,C) we use the same nota-tions and employ matrices similar to those introduced to study so(2r+1,C). Morespecific: For 1≤ j,k ≤ r we extend the matrix

E j,k ∈M(r× r,R)'M(2r×2r,C)

by zero to the matrix

E j,k ∈M((2r+1)× (2r+1),C).

Like E j,k the matrix E j,k has non-zero entries only at the two places (2 j−1,2k−1)and (2 j,2k), both entries with value = 1.

Proposition 7.10 (Typ Br). The Lie algebra L := so(2r+ 1,C),r ≥ 2, has the fol-lowing characteristics:

1. dim L = r(2r+1).

2. The subalgebra

H := spanC < h · E j, j : 1≤ j ≤ r > ⊂ d(2r+1,C)∩L


3. For 1≤ j ≤ r denote byε j := (h · E j, j)

∗ ⊂ H∗

the dual functionals. For any pair 1≤ j < k ≤ r each of the four elements


s · (E j,k− Ek, j), t · (E j,k− Ek, j),u · E j,k− v · Ek, j,v · E j,k−u · Ek, j

generates the 1-dimensional root space belonging to the respective root

α =

ε j + εk

−ε j− εk

ε j− εk

−ε j + εk

In addition, for each index j = 1, ...,r a 1-dimensional root space is generatedby the matrix

X j ∈ so(2r+1,C)

with exactly four non-zero entries: The vector

B1 :=(

i1

)∈M(2×1C)

at places (2 j−1,2r+1) and (2 j,2r+1) and the vector B>1 at places (2r+1,2 j−1)and (2r+1,2 j).

And in addition, for each index j = 1, ...,r a 1-dimensional root space is gener-ated by the matrix

Yj ∈ so(2r+1,C)

with exactly four non-zero entries: The vector

B2 :=(

i−1

)∈M(2×1,C)

at places (2 j−1,2r+1) and (2 j,2r+1) and the vector B>2 at places (2r+1,2 j−1)and (2r+1,2 j).

For j = 1, ...,r the respective roots belonging to X j and Yj are

α =±ε j.

4. The root system Φ of L is

Φ = ±εk± εn : 1≤ k < n≤ r∪±ε j : j = 1, ...,r.

A base of φ is the set∆ = α1, ...,αr

with - note the different form of the last root -

• α j := ε j− ε j+1,1≤ j ≤ r−1,


• αr := εr.

The set of positive roots is

Φ+ = εi± ε j : 1≤ i < j ≤ r∪ε j : 1≤ j ≤ r.

5. For a positive root α ∈Φ+ the subalgebra

Sα =< hα ,xα ,yα >' sl(2,C)

has the generators:

• If α = ε j + εk,1≤ j < k ≤ r, then

hα := h · (E j, j +Ek,k),xα := s · (E j,k−Ek, j),yα := t · (E j,k−Ek, j).

• If α = ε j− εk,1≤ j < k ≤ r, then

hα := h · (E j, j−Ek,k),xα := u ·E j,k− v ·Ek, j,yα := v ·E j,k−u ·Ek, j.

• If α = ε j,1≤ j ≤ r, then

hα := 2h ·E j, j,xα = X j,yα = Yj.

6. The Cartan matrix of the root system Φ referring to the base ∆ is

Cartan(∆) =

2 −1 0 . . . 0−1 2 −1 . . . 0

. . .. . .

. . . 00 . . . 0 −1 2 −20 0 −1 2

∈M(r× r,Z).

Note the distinguished entries in the last row and the last column.The roots α j, j = 1, ...,r− 1, have equal length; they are the long roots. Theroot αr is the single short root. The length ratio is

‖α j‖‖αr‖

=√

2, j = 1, ...,r−1.

The non-orthogonal pairs of roots from ∆ are

(α j,α j+1), j = 1, ...,r−1.

Each pair (α j,α j+1), j = 1, ...,r − 2, encloses the angle2π

3, while the last

pair (αr−1,αr) encloses the angle3π

4. The Dynkin diagram of Φ has type Br

from Theorem 6.26.


Proof. The canonical embedding

so(2l,C) −→ so(2l +1,C)

extends a matrix X ∈ so(2l,C) by adding a zero-column as last column and a zero-row as last row. Therefore most part of the proof follows from the correspondingstatements in Proposition 7.9. In addition:

3) The general element of H has the form

z =r

∑ν=1

aν ·h ·Eν ,ν ,aν ∈ C.

In addition to the action of H on the root space elements from Proposition 7.9 onehas the action on the additional elements X j and Yj: For 1≤ j≤ r the element z ∈Hacts according to

[z,X j] = ε j(z) ·X j, [z,Yj] =−ε j(z) ·Yj.

4) The positive roots have the base representation:

ε j =r

∑k= j

αk,1≤ j ≤ r,

εi− ε j =j−1

∑k=i

αk,1≤ i < j ≤ r.

5) Note:B1 ·B>2 −B2 ·B1

> = 2 ·h ∈M(2×2,C).

6) The computation of the Cartan matrix is similar to the computation in theproof of Proposition 7.9, q.e.d.

Theorem 7.11 (The classical complex Lie algebras are simple). The Lie algebrasof the complex classical groups belonging to types

Ar,r ≥ 1;Br,r ≥ 2;Cr,r ≥ 3;Dr,r ≥ 4,

are simple.

Proof. We know from Proposition 7.6 that any classical Lie algebra L is semisimple.According to Theorem 4.15 the semisimple Lie algebra L splits into a direct sum ofsimple Lie algebras. Elements from different summands commute pairwise. There-fore the direct sum reduces to one single summand, because the Dynkin diagram ofL is connected. Hence L is simple, q.e.d.


Remark 7.12 (Semisimplicity of the classical Lie algebras). Apparently, one can for-mulate the content of Proposition 7.7 until Proposition 7.10 without using that theclassical Lie algebras L are semisimple. Then these propositions give a second prooffor the semisimplicity of L, besides Proposition 7.6. One needs the decompositionof L as a direct sum of H and 1-dimensional eigenspaces Lα

L = H⊕⊕α∈Φ

Lα ,

the existence of the subalgebras

Sα ' sl(2,C)⊂ L

and the resultspanC < Φ >= H∗ :

The radicalR := rad(L)⊂ L

is an ideal. Therefore the subspace R⊂L is ad(H)-invariant. According to Lemma 5.2the eigenspace decomposition of L

L = H⊕⊕α∈Φ

Lα

into eigenspaces of ad(H) induces the vector space decomposition

R = (R∩H)⊕⊕α∈Φ

(R∩Lα).

Assume R 6= 0, then:

i) Claim R 6⊂ H: Assume on the contrary R⊂ H. Then a non-zero element

h ∈ R⊂ H

exists. Because Φ spans the dual space H∗, an element α ∈Φ exists with

α(h) 6= 0.

Consider a non-zero element x ∈ Lα . From h ∈ H follows

[h,x] = α(h) · x.

And from h ∈ R and R⊂ L being an ideal follows

[h,x] =−[x,h] ∈ R⊂ H

which implies x ∈ H. But H ∩Lα = 0, a contradiction.


ii) According to part i) a non-zero element x ∈ R\H exists of the form

x = h+ ∑α∈Φ

xα

satisfying for all α ∈Φ

xα ∈ R∩Lα .

As a consequence for at least one α ∈Φ :

xα 6= 0.

The root space Lα is 1-dimensional.

• If α ∈Φ+ then we consider the subalgebra

spanC < xα ,yα ,hα >' sl(2,C)⊂ L.

We use that R⊂ L is an ideal. From xα ∈ R∩Lα follows

hα = [xα ,yα ] ∈ R

andyα =−(1/2) · [hα ,yα ] ∈ R.

Thereforesl(2,C)' spanC < xα ,yα ,hα >⊂ R,

a contradition, because R is solvable but sl(2,C) is not.

• If α ∈Φ− then we consider the subalgebra

spanC < x−α ,y−α ,h−α >' sl(2,C)⊂ L

and start with yα ∈ R. In an analogous way we obtain

sl(2,C)' spanC < x−α ,y−α ,h−α >⊂ R,

a contradiction.

Both cases i) and ii) lead to a contradition. This fact shows R = 0 and provesthat L is semisimple, q.e.d.

Remark 7.13 (Real simple Lie algebras). Also the Lie algebras of the real classicalgroups belonging to types

Ar,r ≥ 1;Br,r ≥ 2;Cr,r ≥ 3;Dr,r ≥ 4;

are simple.

7.3 Outlook 217

7.3 Outlook

According to Theorem 7.5 the root set of any complex semisimple Lie alge-bra is a root system. The previous section shows that all Dynkin diagrams fromthe A,B,C,D-series, see Theorem 6.26, are realized by the root system of a simpleLie algebra.

At least the following related issues remain:

• Show that also the exceptional Dynkin diagrams of type F4,G2,Er with r = 6,7,8result from root systems, see [28, Chap. 12.1].

• Show that also the exceptional root systems can be obtained from simple Liealgebras.

• Show that a semisimple Lie algebra is uniquely determined by its root system.

Chapter 8Irreducible Representations

Representation theory of complex semisimple Lie algebras L rests on the followingresults from the general theory:

• The Cartan decomposition of L and the corresponding root system.

• Weyl’s theorem on complete reducibility of finite dimensional L-modules.

• Representation theory of the particular Lie algebra sl(2,C).

• The universal enveloping algebra of a Lie algebra.

8.1 The universal enveloping algebra

In the present section L denotes an arbitrary K-Lie algebra, not necessaritly semisim-ple. The Lie algebra and the vector spaces may have infinite dimension if not statedotherwise.

Before studying the structure of L-modules we make a disgression from Lie al-gebras to associative algebras. We assume that all associative algebras A have a unitelement 1 ∈ A and all morphism ρ between associative algebras respect the unitelement.

We introduce the universal enveloping algebra UL of a Lie algebra L.

Any associative K-algebra A defines a K-Lie algebra with Lie bracket

[x,y] := x · y− y · x, x,y ∈ A.

219

220 8 Irreducible Representations

Conversely any Lie algebra L embedds into an associative algebra UL by a universalconstruction. Even for finite-dimensional L the universal enveloping algebra UL isnot necessarily finite dimensional.

Definition 8.1 (Tensor algebra, symmetric algebra and universal enveloping al-gebra).

1. Consider a K-vector space M. The tensor algebra T M of M is the pair (T M, j)with

• the associative K-algebra

T M :=⊕k∈N

T kM, T kM := M⊗k,

with M0 =K, unit 1∈K, and the algebra multiplication induced by the canon-ical map

T mM×T nM→ T m+nM,(u,v) 7→ u⊗v,

• and the canonical K-linear injection

j : M = T 1M → T M.

2. For a K-vector space M denote by

I := span < m1⊗m2−m2⊗m1 : m1, m2 ∈M >⊂ T M

the two sided ideal generated by all symmetric tensors. Set

SM := T M/I

and denote bypSM : T M −→ SM

the canonical projection. The pair

(SM, pSM j)

is named the symmetric algebra of M.

3. Consider a K-Lie algebra L. The universal enveloping algebra of L is thepair (UL, i) with

• The associative algebra UL obtained from the tensor algebra T L as the quo-tient

UL := T L/J

with J ⊂ T L defined as the two-sided ideal generated by all elements

j([a,b])− ( j(a)⊗ j(b)− j(b)⊗ j(a)), a,b ∈ L,

8.1 The universal enveloping algebra 221

• and the K-linear map

i : L→UL, x 7→ pUL( j(x)),

withpUL : T L→UL

the canonical projection.

The associative product in UL is denoted

xy ∈UL or x · y ∈UL for x, y ∈UL.

Moreover we consider the map j : L −→ T L as an identification. The restrictionon K= T 0L

pUL|T 0L : T 0L−→UL

is injective, becauseI ⊂

⊕m≥1

L⊗m.

A deeper fact states that also the map

i : L−→UL

embeds L into UL, see Proposition 8.9, part 2).

In the specific case of an Abelian Lie algebra L the ideal I ⊂ T L is generated byall elements of the form

u⊗v−v⊗u with u,v ∈ T M,

and the universal enveloping algebra UL equals the symmetric algebra

SL := T L/span < u⊗v−v⊗u : u, v ∈ L > .

For a finite-dimensional Lie algebra L the symmetric algebra is isomorphic to thepolynomial algebra

UL'K[X1, ...,Xn], n = dim L.

The name universal enveloping algebra indicates that the pair (UL, i) has a uni-versal property:

Lemma 8.2 (Universal property). Consider a K-Lie algebra L. For any associative K-algebra Aand any K-linear morphism ρ : L→ A satisfying

ρ([a,b]) = ρ(a) ·ρ(b)−ρ(b) ·ρ(a)

exists a unique morphism of K-algebras


ρUL : UL→ A

such that the following diagram commutes:

UL

L

A

i

ρ

ρUL

Proof. Define ρUL : UL−→ A by

ρUL(pUL(x1⊗ ...xn)) := ρ(x1) · ... ·ρ(xn), q.e.d.

Often the study of an associative algebra A can be facilitated by choosing a filtra-tion, i.e. a suitable exhaustion of the algebra by subspaces, such that the successivequotients make up a graded algebra GA. The latter algebra can often be investi-gated more easily than the original algebra A. The graded algebra GA is similar to apolynomial algebra, which splits as a direct sum of the subspaces of homogeneouspolynomials of fixed degree. The Poincare-Birkhoff-Witt Theorem 8.8 provides anexample for the case of the universal enveloping algebra A =UL of a Lie algebra L.

For a fixed Lie algebra L we use the shorthand

T := T L, T⊗m := T Lm, S := SL, U :=UL.

Definition 8.3 (Filtration and graded algebra of the universal enveloping alge-bra). Consider an arbitrary but fixed K-Lie algebra L with universal envelopingalgebra U .

1. For m ∈ N denote by

FmU := spanK < x1 · ... · xi ∈ L : 1≤ i≤ m >⊂U

the K-subspace of the universal enveloping algebra formed by all products withat most m factors from L. In particular,

F0U =K and F1U =K+L.

ThenFU := (FmU)m∈N

is a filtration of the universal enveloping algebra U :

K= F0U ⊂ F1U ⊂ ...⊂ FmU ⊂ ...


satisfyingFnU ·FmU ⊂ Fn+mU.

2. The successive quotient vector spaces of the filtration

GmU := FmU/Fm+1U, m ∈ N,

define the commutative associative algebra

GU :=⊕m∈N

GmU

with multiplication on the summands defined as follows: Set

GnU := FnU/Fn−1U, GmU := FmU/Fm−1U, Gn+mU := Fn+mU/Fn+m−1U.

Then

GnU×GmU → Gn+mU, (x+Fn−1U) · (y+Fm−1U) := x · y+Fn+m−1U.

There is a canonical morphism of algebras

pU,GU : U −→ GU

defined as follows: For a non-zero element x∈U choose the minimal index m∈Nsatisfying

x ∈ FmU

and definepU,GU (x) := x+Fm−1U ∈ GmU.

The pair (GU, pU,GU ) is named the graded algebra of the universal envelopingalgebra of L.

3. In general the Lie algebra L is not Abelian. For non-Abelian L the ideal

J := span < [a,b]− (a⊗b−b⊗a) : a,b ∈ L >⊂ T

is not homogeneous. As a consequence the grading of the tensor algebra T doesnot induce a grading of the quotient U = T/J by homogeneous elements. Never-theless, the filtration (Tm)m∈N of T induces a filtration

(FmU := pU (Tm))m∈N

of U . Taking the associated filtration defines the graded algebra

GU :=⊕m∈N

GmU

and a canonical morphism


pU,GU : U −→ GU.

4. The notation FmU denotes the filtration by lower indices, while the notation GmUdenotes the graduation by upper indices.

Remark 8.4 (Graded universal enveloping algebra and symmetric algebra).

1. Multiplication: The multiplication of the graded algebra GU from Definition 8.3is well-defined because for x ∈ Fn−1U or y ∈ Fm−1U

x · y ∈ Fn+m−1U.

2. Commutativity and relation to symmetric algebra: The graded algebra GU iscommutative: Consider the composition

pGU := [pU,GU : U −→ GU ] [pU : T −→U ] : T −→ GU

and the ideal

I := span < v1⊗v2−v2⊗v1 : v1, v2 ∈ L >⊂ T.

On has for all v1,v2 ∈ L

pU (v1⊗v2−v2⊗v1) = pU ([v1,v2]) ∈ F1U.

HencepGU (I)⊂ 0= F1U/F1U ⊂ F2U/F1U.

The inclusion I ⊂ ker pGU implies the existence of a morphism

φ : SL−→ GU

such that the following diagram commutes

T GU

S

pGU

pS φ

Surjectivity of implies that φ is surjective. The commutativity of S implies thecommutativity of

GU = φ(S).

The main theorem about the universal enveloping algebra UL is due to Poincare-Birkhoff-Witt, see Theorem 8.8. It states that φ is an isomorphism of associative


algebras. Hence the graded algebra GUL of the universal enveloping algebra ULis the algebra of polynomials in finitely many variables, a well-understood alge-braic object. The proof of the injectivity of φ in requires some preparations, seeNotation 8.5, Proposition 8.6 and Lemma 8.7.

Notation 8.5. Denote by L a K-Lie algebra and by SL its symmetric algebra. Weassume a basis of L

xλ : λ ∈Ω

with an ordered index set Ω . For any λ ∈ Ω we denote by zλ a correspondingindeterminate for the polynomial algebra

SL'K[(zλ )λ∈Ω ].

For any m ∈N we denote the subspace of homogeneous polynomials of degree m by

SmL := spanK < zλ1 · ... · zλm >⊂ SL.

These subspaces define the graded algebra

SL =⊔

m∈NSmL.

It results from the filtration

SmL := S0L⊕ ...⊕SmL⊂ SL, m ∈ N,

by the subspaces of polynomials of degree ≤ m. For a sequence

Σ = (λ1, ...,λm)

we setxΣ := xλ1 ⊗ ...⊗ xλm ∈ T mL

andzΣ := zλ1 · ... · zλm ∈ SmL.

If λ1 ≤ ...≤ λm then the sequence Σ is increasing.

Proposition 8.6 (SL as L-module). Consider a K-Lie algebra L with symmetricalgebra SL. Then a unique morphism of Lie algebras exists

ρ : L−→ gl(SL)

satisfying for all λ ≤ Σ ,ρ(xλ )zΣ = zλ · zΣ ,

and for all m ∈ N and all Σ of length m


ρ(xλ )zΣ ≡ zλ · zΣ mod Sm.L

The algebraic proof of the proposition follows a standard scheme, see [28, § 17.4],[31, Ch. II.6], [44, Part I, Ch. III, No. 4] .

Proof. Because the K-algebra SL is Abelian we may assume all m-tuples in theproof

Σ = (λ1, ...,λm)

as increasing.

i) The existence of the representation ρ is equivalent to the existence of a linearmap

f : L⊗SL−→ SL,

which satisfies for all z1,z2 ∈ L and all v ∈ SL

f ([z1,z2]⊗v) = f (z1⊗ f (z2⊗v))− f (z2⊗ f (z1⊗v)).

Due to the filtration of SL the existence of the linear map f is equivalent to a com-patible family ( fm)m∈N of linear maps

fm : L⊗SmL−→ GS,

which satisfy for all m-tuples Σ the following properties:

1. Am: For all λ ≤ Σ and all zΣ ∈ SmL

fm(xλ ⊗ zΣ ) = zλ · zΣ ∈ Sm+1L

2. Bm: For all λ ∈Ω , for all k ≤ m and for all zΣ ∈ SkL

fm(xλ ⊗ zΣ )≡ zλ · zΣ mod SkL

3. Cm: For all λ , µ ∈Ω , for all m−1-tuples T and for all zT ∈ Sm−1L

fm([xλ ,xµ ]⊗ zT ) = fm(xλ ⊗ fm(xµ ⊗ zT ))− fm(xµ ⊗ fm(xλ ⊗ zT )).

Here condition Am considers for given Σ only indices λ ∈ Ω less or equal thaneach index from Σ . Condition Bm considers arbitrary indices λ and allows a correc-tion term of lower order in comparison to the result from Am. While condition Cmcaptures the requirement for commutators from L.

ii) We construct the family f := ( fm)m∈N by induction on m.

Induction start m = 0: The only sequence to consider is the empty sequence Σ = /0with zΣ = 1. Moreover

S0L = S0L =K.


We define for all λ ∈Ω

f0 : L⊗S0L−→ SG, f0(xλ ⊗1) := zλ ∈ S1L.

Apparently f0 is the only linear map which satisfies condition A0. It also satisfiescondition B0 and C0.

Induction step m−1 7→ m: We have to extend fm−1, defined on L⊗Sm−1L, to alinear map fm defined on L⊗SmL. Therefore we have to define

fm(xλ ⊗ zΣ ) ∈ SL

for an increasing sequence Σ of lenght = m.

If λ ≤ Σ then property Am requires the definition

fm(xλ ⊗ zΣ ) := zλ · zΣ .

Otherwise, Σ = (λ1, ...,λm) satisfies

λ > µ := λ1.

We set Σ = (µ,T ) with the increasing (m−1)-tuple T = (λ2, ...,λm).Because µ ≤ T , the induction hypothesis Am−1 implies

fm−1(xµ ⊗ zT ) = zµ · zT = zΣ .

The requirement of compatibility between fm and fm−1 requires the definition

fm(xµ ⊗ zT ) := fm−1(xµ ⊗ zT ) = zΣ .

As a consequence, condition Cm requires

fm([xλ ,xµ ]⊗ zT ) = fm(xλ ⊗ zΣ )− fm(xµ ⊗ fm(xλ ⊗ zT ))

orfm(xλ ⊗ zΣ ) = fm(xµ ⊗ fm(xλ ⊗ zT ))+ fm([xλ ,xµ ]⊗ zT ).

Concerning the terms on the right-hand side of the last equation:

• First term: The compatibility of fm with fm−1 and the inductionhypothesis Bm−1 imply

fm(xλ ⊗ zT ) = fm−1(xλ ⊗ zT )≡ zλ · zT mod Sm−1L

orfm(xλ ⊗ zT ) = zλ · zT + y,

withy = fm−1(xλ ⊗ zT )− zλ · zT ∈ Sm−1L.


• Second term: The compatibility of fm with fm−1 implies

fm([xλ ,xµ ]⊗ zT ) = fm−1([xλ ,xµ ]⊗ zT ).

The compatibility of fm with fm−1 and condition Am imply

fm(xµ ⊗ fm(xλ ⊗ zT )) = fm(xµ ⊗ zλ · zT )+ fm(xµ ⊗ y) =

= fm(xµ ⊗ zλ · zT )+ fm−1(xµ ⊗ y) = zµ · zλ · zT + fm−1(xµ ⊗ y).

Summing up: We must define

fm(xλ ⊗ zΣ ) := zµ · zλ · zT + fm−1(xµ ⊗ y).

After having now defined as extension of fm−1 the linear map

fm : L⊗SmL−→ Sm+1L

in accordance with condition Am and Bm, it remains to show that fm satisfiescondition Cm. Hence consider arbitrary but fixed

λ , µ and an increasing (m−1)-index tuple T.

Condition Cm has already been verified if

µ < λ and µ ≤ T.

Condition Cm also holds for the reverse case

λ < µ and λ ≤ T

because[xµ ,xλ ] =−[xλ ,xµ ].

Apparently, condition Cm holds for λ = µ . Hence we are left with the case:

neither λ ≤ T nor µ ≤ T.

Because we now fix m, we introduce the shorthand (dot on the line)

x.z := fm(x⊗ z)

Decomposing T = (ν ,Ψ) with lenght Ψ = m−2. Then

λ > ν and µ > ν .

We obtainxµ .zT = xµ .(zν · zΨ )


using condition Am−2 for the last equality. Condition Cm−1 applied with theindices µ and ν implies

xµ .(xν .zΨ ) = xν .(xµ .zΨ )+ [xµ ,xν ].zΨ ,

hencexµ .zT = xν .(xµ .zΨ )+ [xµ ,xν ].zΨ .

Condition Bm−2, a consequence of Bm−1, implies the existence of asuitable w ∈ Sm−2L such that

xµ .zΨ = zµ · zΨ +w.

Hencexµ .zT = xν .(zµ · zΨ )+ xν .w+[xµ ,xnu].zΨ .

We apply xλ to both sides of the equation:

xλ .(xµ .zT ) = xλ .(xν .(zµ · zΨ ))+ xλ .(xν .w)+ xλ .([xµ ,xnu].zΨ .)

The trick of the proof is now to apply the induction assumption and the part ofcondition Cm, which has already been proven, to each of the three summandsseparately. And afterwards to add the results and to remove the intermediate term wfrom the result. Using the shorthand T1,T2,T3 for the three summands on theright-hand side of the equation, we transform each of the summands as

• T1: Condition Cm applies because ν ≤ (µ,Ψ), hence

xλ .(xν .(zµ · zΨ )) = xν .(xλ .(zµ · zΨ ))+ [xλ ,xν ].(zµ · zΨ ).

• T2: Condition Cm−1 applies, hence

xλ .(xν .w) = xν .(xλ .w)+ [xλ ,xν ].w

• T3: Condition Cm−1 applies, hence

xλ .([xµ ,xν ].zΨ ) = [xµ ,xν ].(xλ .zΨ )+ [xλ , [xµ ,xν ]].zΨ

Summing up the three summands and combining the corresponding terms of T1and T2 as

xν .(xλ .(zµ · zΨ ))+ xν .(xλ .w) = xν .(xλ .(xµ .zΨ ))

[xλ ,xν ].(zµ · zΨ )+ [xλ ,xν ].w = [xλ ,xν ].(xµ .zΨ )

we obtain as a first equation

xλ .(xµ .zT )= xν .(xλ .(xµ .zΨ ))+[xλ ,xν ].(xµ .zΨ )+[xµ ,xν ].(xλ .zΨ )+[xλ , [xµ ,xν ]].zΨ

We now interchange the role of λ and µ and subtract the resulting equation fromthe first equation. Due to the symmetry the underlined terms from both equations


cancel. The result completes the induction step:

xλ .(xµ .zT )− xµ .(xλ .zT ) =

= xν .(xλ .(xµ .zΨ ))− xν .(xµ .(xλ .zΨ ))+ [xλ , [xµ ,xν ]].zΨ − [xµ , [xλ ,xν ]].zΨ =

= xν .([xλ ,xµ ].zΨ )+ [xλ , [xµ ,xν ]].zΨ +[xµ , [xν ,xλ ]].zΨ =

= [xλ ,xµ ].(xν .zΨ )+ [xν , [xλ ,xµ ]].zΨ +[xλ , [xµ ,xν ]].zΨ +[xµ , [xν ,xλ ]].zΨ =

= [xλ ,xµ ].zT +([xν , [xλ ,xµ ]]+ [xλ , [xµ ,xν ]]+ [xµ , [xν ,xλ ]]).zΨ =

= [xλ ,xµ ].zT

Here the last equality is due to the Jacobi identity. The end of the induction stepfinishes the proof of the Proposition, q.e.d.

Lemma 8.7 (Commutator versus symmetric element). Denote by L a Lie algebrawith universal enveloping algebra UL and set

I := ker [pSL : T L−→ SL], J := ker [pUL : T L−→UL].

Then for any elementt ∈ T mL∩ J

of degree at most = m the homogeneous component of degree = m belongs to I

tm ∈ T mL∩ I.

The meaning of Lemma 8.7 can be easily seen in the case m = 2 by the element

t = v1⊗v2−v2⊗v1− [v1,v2] ∈ T 2L∩ J :

Apparently its homogeneous part of degree = 2 is symmetric

t2 = v1⊗v2−v2⊗v1 ∈ I.

Proof. i) The morphism of Lie algebras

ρ : L−→ gl(SL)

from Proposition 8.6 can be considered as a linear map

ρ : L−→ End(SL)

satisfyingρ([x,y]) = ρ(x)ρ(y)−ρ(y)ρ(x).

Due to the universal property of the universal enveloping algebra UL we obtain aunique morphism of associative algebras


ρUL : UL−→ End(SL)

such that the following diagram commutes

L

T L End(SL)

UL

ρj

pULρUL

We consider the morphism of associative algebras

ρ := ρUL pUL : T L−→ End(SL)

satisfying ρ(1) = 1. Apparently J ⊂ ker ρ .

ii) For t ∈ T mL the endomorphism

ρ(t) ∈ End (SL)

maps the unit to the elementρ(t)(1) ∈ SL.

Consider an orderded basis (xλ )λ∈Ω of L. For each sequence Σ = (λ1, ...,λk)with xΣ a summand of t we apply Proposition 8.6 successively for λk, ...,λ1 andobtain

ρ(xΣ )(1) = zΣ ·1 mod Sk−1L = zΣ mod Sk−1L.

Henceρ(t)(1) = ∑

Σ

zΣ mod Sm−1L.

If also t ∈ J, then its homogeneous component of highest degree is a symmetrictensor:

tm = ∑length Σ=m

xΣ

ortm ∈ I∩T mL, q.e.d.

For an Abelian Lie algebra L holds

SL'UL.


The following Theorem 8.8 shows how to generalize this fact to arbitrary Liealgebras.

Theorem 8.8 (Poincare-Birkhoff-Witt). For any K-Lie algebra L the canonicalmorphism

φ : SL→ GUL

from the symmetric algebra of L to the graded algebra GUL of the universal en-veloping algebra UL is an isomorphism of associative K-algebras.

Proof. 1. Surjectivity. The morphism pGUL : T L −→ GUL is surjective, because foreach m ∈ N the map

L⊗m −→ FmU/Fm−1U

is surjective. Hence the induced map φ is surjective.

2. Injectivity: We setI := ker pSL ⊂ T L

in the commutative diagram

T L GUL

SL

pSL

pGUL

φ

Moreover we recall the canonical map

pUL : T L−→UL

and setJ := ker pUL.

We show for any m ∈ N: If t ∈ T mL satisfies

pGUL(t) = 0

i.e.pUL(t) ∈ Fm−1UL

then t ∈ I, hence pSL(t) = 0. The particular case m= 1 proves the injectivity of φ .

Consider a homogeneous element t ∈ T mL of degree =m with pUL(t)∈Fm−1UL.Then an element t ′ ∈ Tm−1L of degree at most = m−1 exists with

pUL(t) = pUL(t ′)


i.e.pUL(t− t ′) = 0 or t− t ′ ∈ J.

Lemma 8.7, applied to the element

t− t ′ ∈ T mL∩ J

with t its homogeneous component of degree = m, implies t ∈ I, q.e.d.

The following Proposition 8.9 follows from Theorem 8.8. It states in particularthat any Lie algebra L embeds into its universal enveloping algebra UL.

Proposition 8.9 (Basis of the universal enveloping algebra). Consider a finite-dimensional Lie algebra L with tensor algebra T , symmetric algebra S, universalenveloping algebra U, and graded universal enveloping algebra GU.

1. If W ⊂ T m is a subspace which under the restriction of the canonical map

pS : T −→ S

is mapped isomorphically to Sm, then

pU (W )⊂ FmU

is a complement to the subspace Fm−1U ⊂ FmU.

2. The canonical K-linear map

i = pU j : L−→U

is injective.

3. Consider a basis (xk)k=1,...,n of L. Then the family of elements

xk1 · ... · xkm := pU ( j(xk1)⊗ ...⊗ j(xkm)) ∈U, m ∈ N and k1 ≤ ...≤ km,

together with the element 1 is a basis of U. Note that the factors are arrangedaccording to increasing indices of the basis elements.

4. Consider a subalgebra H ⊂ L. Extend a basis BH of H to a basis

BL =BH ∪BL/H , BL/H = (xk)k=1,...,n,

of L. Hereby all elements from BH are to be placed before the elements from BL/H .Then the injection H → L induces an injection

UH →UL,

and UL is a free UH-module with basis


xk1 · ... · xkm ,m ∈ N and k1 ≤ ...≤ km,

together with the element 1.

Proof. 1. In the commutative diagram

T U

S GU

pU

pU,GU

φ

the map φ is an isomorphism due to theorem 8.8. The diagram induces by re-striction for each index m ∈ N the commutative diagram

T m FmU

Sm GmU'

By assumption the composition

W −→ T m −→ Sm −→ GmU

is an isomorphism. Hence also the composition

W −→ T m −→ FmU −→ GmU

is an isomorphism, which implies

pU (W )/(pU (W )∩Fm−1U) = pU,GU (W )' GmU = FmU/Fm−1U.

BecausepU (W )∩Fm−1U = 0

we obtainpU (W )' FmU/Fm−1U

orFmU ' pU (W )⊕Fm−1U.

2. We apply the result from part 1) with index m := 1 and W := j(L). Then

pU ( j(L))⊕F0U ' F1U = L⊕K

which impliesi(L) = pU ( j(L))' L.


3. We apply the result of part 1) for a given index m ∈ N and the subspace

W := span⟨xk1 ⊗ ...⊗ xkm : k1 ≤ ...≤ km

⟩⊂ T m.

We obtain that pU (W )⊂ FmU is a complement of Fm−1U ⊂ FmU i.e.

FmU = Fm−1U⊕ pU (W ).

By induction on m ∈ N we build the basis of U .

4. According to part 3) the basis representation with coefficients from K of an ele-ment z ∈UL by the basis derived from BL can be considered a basis representa-tion of x with coefficients from UH by the basis derived from BL/H , q.e.d.

All statements from Proposition 8.9 hold also for Lie algebras with a countableinfinite dimension. The proof goes through along the same lines.

Example 8.10 (Basis of the universal enveloping algebra). Consider the Lie algebra

L = sl(2,C).

The family (1,h,x,y) induces the basis of UL with the elements

hix jyk, i, j,k ∈ N.

The following table shows the product formulas which allow to compute inductivelythe product of two basis elements. The entries of the table show the product with anelement from the left column with an element from the upper row.

h x yh h2 hx hyx hx−2x x2 xyy hy+2y xy−h y2

A consequence of Lemma 8.2 is the following Proposition 8.11.

Proposition 8.11 (The categories of associative algebras and Lie algebras re-spectively). The functor

Lie : Ass−→ Lie

from the category Ass of associative K-algebras to the category Lie of K-Lie alge-bras has a left-adjoint functor


U : Lie−→ Ass,

i.e. for any pair of a K-Lie algebra L and an associative K-algebra A there is anatural isomophism of K-vector spaces

HomLie(L,Lie(A))−→ HomAss(UL,A).

Corollary 8.12 (Equivalence of categories). For any Lie algebra L the category L-modof L-modules and the category UL-mod of UL-modules are equivalent.

8.2 General weight theory

The present section generalizes the representation theory of the Lie algebra sl(2,C)from Chapter 5.2 to the representation theory of an arbitrary complex semisimpleLie algebra L. The L-modules M under consideration are not necessarily finite di-mensional.

We fix

• a pair (L,H) with a semisimple complex Lie algebra L and a Cartan subalgebra H ⊂ L,

• and a base∆ = α1, ...,αr

of the roots Φ of (L,H).

Concerning the corresponding root system Φ we denote

• the decomposition into respectively positive and negative roots by

Φ = Φ+ ∪ Φ

−,

• half the sum of the positive roots by

δ = (1/2) · ∑α∈Φ+

α,

• and the Weyl group of Φ by W .

We recall the Cartan decomposition of L from Definition 5.17

L = H⊕⊕α∈Φ

Lα = H⊕⊕

α∈Φ+

(Lα ⊕L−α).

8.2 General weight theory 237

The subalgebra of LB := H⊕

⊕α∈Φ+

Lα

which comprises besides H only the root spaces of positive roots, is named thecorresponding Borel subalgebra. Moreover we consider the subalgebras of L ofroot spaces belonging to respectively positive and negative roots

N+ :=⊕

α∈Φ+

Lα , N− :=⊕

α∈Φ−Lα .

According to Proposition 7.3 we choose for each positive root α ∈Φ+ a set ofdistinguished elements

xα ∈ Lα , yα ∈ L−α , hα = [xα ,yα ] ∈ H

satisfyingspanC < hα ,xα ,yα >' sl(2,C).

Proposition 8.13 (Solvability of Borel subalgebras). Consider a semisimple Liealgebra L with Cartan subalgebra H. Then the corresponding Borel subalgebraB⊂ L is solvable, and the subalgebras N+, N− ⊂ L are nilpotent.

Proof. The algebra N+ is nilpotent because its descending central series convergesto zero: One uses the formula

[Lα1 ,Lα2 ]⊂ Lα1+α2 .

The same argument shows the nilpotency of N−.

The Borel algebra B is solvable because its commutator algebra

[B,B]⊂⊕

α∈Φ+

Lα = N+

is nilpotent, q.e.d.

Both, the terms in Definition 5.5 from the representation theory of sl(2,C)-modulesand the terms in Definition 5.11 for the adjoint representation of semisimple com-plex Lie algebras generalize to representations of arbitray semisimple complex Liealgebras.

Definition 8.14 (Weight and weight space). Consider a semisimple Lie algebra Land an L-module V .


1. For a linear functional µ ∈ H∗ denote by

V µ := v ∈V : h.v = µ(h) ·v f or all h ∈ H

the eigenspace with respect to the action of H. If V µ 6= 0 then V µ is named aweight space of V , the corresponding linear functional µ ∈ H∗ is a weight of V ,and the non-zero elements of V µ are named weight vectors.

2. A weight µ ∈ H∗ is integral iff for all α ∈ ∆

µ(hα) ∈ Z,

it is non-negative iff for all α ∈ ∆

µ(hα)≥ 0.

Lemma 8.15 (Action of the root spaces). Consider a semisimple Lie algebra L andan L-module V .

1. For any root α of L and any weight µ of V we have

Lα .V µ ⊂V α+µ .

2. The sumV ′ := ∑

µ weightV µ

is direct, i.e.V ′ =

⊕µ weight

V µ .

It is a submodule of V .

Proof. 1. The proof is analogous to the proof of Lemma 5.12.

2. Elements of the Cartan subalgebra H ⊂ L act as semi-simple endomorphims ofV which pairwise commute. The claim follows from the fact that eigenvectorsbelonging to different eigenvalues are linearly independent. The Cartan decom-position of L shows that V ′ ⊂V is a submodule, q.e.d.

Consider a vector space V . Then Lemma 8.2 applies to the associative algebra A = gl(V ):Any representation

ρ : L→ gl(V )

extends uniquely to a morphism of associative algebras

ρUL : UL→ gl(V ).


Analogously to Definition 4.16, we often use the shorthand

u.v := ρUL(u)(v)

for u ∈UL and v ∈V.

Any L-module V with L a complex semisimple Liealgebra is in particular a mod-ule over the solvable Borel subalgebra B ⊂ L. If L is finite-dimensional, then Lie’stheorem applies: The action of B on V has a common eigenvector. Such an eigen-vector is called a primitive element of V .

Definition 8.16 (Primitive element and standard cyclic module). Consider acomplex semisimple Lie algebra L with Borel subalgebra B⊂ L, and an L-module V .

1. A weight vector e ∈V λ is named a primitive element or maximal vector of V if

b.e = λ (b) · e for all b ∈ B.

2. An L-module V is standard cyclic if it has a primitive element e ∈ V whichgenerates V via the action of UL, i.e. if

V =UL.e

Lemma 8.17 (Existence of primitive elements). Any finite-dimensional L-module V 6= 0has a primitive element e.

Proof. The Borel algebra B ⊂ L is solvable due to Proposition 8.13. According toTheorem 3.20 a non-zero element e ∈V exists such that for all x ∈ B

x.e = λ (x) · e.

If α ∈Φ+ andxα ∈ Lα ⊂ N+

we considerSα := span < hα ,xα ,yα >' sl(2,C).

Then2xα .e = [hα ,xα ].e = λ (hα) ·λ (xα) · e−λ (xα) ·λ (hα) · e = 0.

HenceN+.e = 0

and e is a primitive element, q.e.d.

The following Proposition 8.18 clarifies the structure of standard cyclic modules;compare also Proposition 5.6.


Proposition 8.18 (Standard cyclic module). Consider a semisimple Lie algebra Lwith basis

∆ = α1, ...,αr

of its root system Φ , and a standard cyclic L-module V with primitive element e ∈V λ .Then:

1. Generation by UN−:V =UN−.e

2. All weights of V have the form

µ = λ −r

∑i=1

mi ·αi, mi ∈ N f or all i = 1, ...,r;

hence the weight λ is named the highest weight of V .

3. Finite dimensionality of weight spaces: For each weight µ of V the weight spaceis finite-dimensional, i.e.

dim V µ < ∞.

In particulardim V λ = 1.

4. The L-module V does not decompose as a direct sum of proper L-submodules,i.e. there is no representation

V =V1⊕V2

with non-zero L-modules V1, V2.

Proof. Note that V is not necessarily finite dimensional: In general there is no lowerbound for the weights µ .

1. Proposition 8.9, part 4) applied to the vector space decomposition

L = N−⊕B

shows: Extending the basis of N−

BN− = (yα)α∈Φ+

by the basis of BBB = (hα)α∈∆ ∪ (xα)α∈Φ+

to a basis of L represents UL as a free UN−-module with basis BB. Therefore

UL.e = (UN− ·UB).e =UN−.e

2. The claim follows from part 1):


Lα .V µ ⊂V µ+α

and the fact, that any negative root α ∈ Φ− is a linear combination of elementsfrom ∆ with negative integer coefficients.

3. According to Proposition 8.9, part 3) the family(y j(1)

α1 · ... · yj(r)αr

)j(1),..., j(r)∈N

of monomials is a basis of UN−. For fixed weight

µ := λ −r

∑i=1

mi ·αi

only finitely many of these monomials satisfy

y j(1)α1 · ... · y

j(r)αr .e ∈V µ .

Hence dim V µ < ∞.

The only monomial generator with

y j(1)α1 · ... · y

j(r)αr .e ∈V λ

is1 = y0

α1· ... · y0

αr ,

hence dim V λ = 1.

4. If V =V1⊕V2 then the weight space of V λ decomposes as

V λ =V1λ ⊕V2

λ .

According to part 3) we have dim V λ = 1, hence w.l.o.g.

dim V1λ = 1 and dim V2

λ = 0.

Therefore e ∈ V1. Because e generates V we conclude V ⊂ V1, i.e. V = V1and V2 = 0, q.e.d.

It is a consequence of Proposition 8.18, that an irreducible L-module with a prim-itive element is standard cyclic. Hence its structure is well-known. Thanks to Weyl’stheorem on complete reducibility also the structure of arbitrary finite-dimensional L-modulesis determined.


Corollary 8.19 (Structure of irreducible standard cyclic modules). Consider asemisimple Lie algebra L and an irreducible L-module V with a primitive element e ∈V λ .Then:

1. The module V is standard cyclic. In particular, all weights of V have the form

µ = λ −r

∑i=1

mi ·αi, mi ∈ N f or all i = 1, ...,r.

Each weight space is finite dimensional; in particular dim V λ = 1. Moreover Vsplits as a direct sum of its weight spaces

V =⊕

µ weight

V µ .

2. Up to scalar multiplication e is the only primitive element of V .

3. Two irreducible L-modules Vi with highest weight

λi, i = 1,2,

are isomorphic iff λ1 = λ2.

Proof. 1. The standard cyclic submodule E ⊂ V λ generated by the primitive ele-ment e ∈ V equals V because the latter is irreducible. Hence Proposition 8.18proves the claim.

2. Assume a second primitive element e′ ∈V . According to part 1) its weight λ ′ hasthe form

λ′ = λ −

r

∑i=1

mi ·αi, mi ∈ N,

Exchanging the roles of e and e′ shows

λ = λ′−

r

∑i=1

mi′ ·αi, mi

′ ∈ N,

Hence mi = mi′ for all i = 1, ...,r and λ = λ ′. Because

dim V λ = 1

according to part 1), the two primitive elements e and e′ are scalar multiples.

3. i) Consider two irreducible L-modules V1 and V2 with the same heighest weight λ .We choose primitive elements e1 ∈V1 and e2 ∈V2 and define the L-module

V :=V1⊕V2

with canonical L-linear projections


pi : V →Vi, i = 1,2.

Then V has a primitive element e = e1 + e2. It generates a submodule E ⊂ V .We have e1 /∈ E because all primitive elements of E are scalar multiples of eaccording to Proposition 8.18. Analogously, e2 /∈ E.

We have p2(e) = e2. Because e2 generates V2 we conclude that

p2|E : E −→V2

is surjective. Its kernel N2 := ker (p2|E) has the form

N2 = E ∩V1.

As a submodule of the irreducible module V1 the module N2 is either zero orequal to V1. The latter equality is impossible because e1 /∈ N2. Hence p2|E isinjective, and therefore an isomorphism

p2 : E '−→V2.

A similar consideration shows that

p1 : E −→V1

is an isomorphism. Therefore V1 'V2.

ii) Consider an isomorphismφ : V1

'−→V2

between two irreducible L-modules. According to part 1) each of the two L-moduleshas a uniquely determined 1-dimensional subspace of primitive elements

ei ∈V λii , i = 1,2.

We may assume φ(e1) = e2. For arbitrary elements h ∈ H from the Cartan sub-algebra H ⊂ L we have

λ2(h) ·e2 = h.e2 = h.φ(e1) = φ(h.e1) = φ(λ1(h) ·e1) = λ1(h) ·φ(e1) = λ1(h) ·e2,

hence λ1(h) = λ2(h). As a consequence

λ1 = λ2 ∈ H∗,q.e.d.

Not any irreducible L-module has a primitive element as the following exampleshows.

Example 8.20 (An irreducible sl(2,C)-module without primitive elements). Con-sider the Lie algebra


L := sl(2,C) = spanC < h,x,y >

with the commutators

[h,x] = 2x, [h,y] =−2y, [x,y] = h.

Similarly to Proposition 5.6 we consider the infinite-dimensional vector space

V := spanC < en : n ∈ 2Z>

with the basis (en)n∈2Z. We define the following action of L using x as raising oper-ator and y as lowering operator:

h.en := n · en

x.en := an · en+2

y.en := bn · en−2.

i) The sequences (an)n∈N, (bn)n∈N have to be choosen in a suitable way: Threeconditions have to be satisfied, in order that these definitions determine an L-modulestructure on V . The first two conditions are fulfilled for any choice: First,

[h,x].en := h.(x.en)− x.(h.en) = 2 ·an · en+2

and2x.en = 2 ·an · en+2.

Secondly[h,y].en := h.(y.en)− y.(h.en) = 2 ·bn · en−2

and2y.en = 2 ·bn · en−2.

The third condition, the equality of

[x,y].en = x.(y.en)−y.(x.en) = x.(bn ·en−2)−y.(an ·en+2) = (an−2 ·bn−an ·bn+2)en

andh.en = n · en,

requires for all n ∈ 2Zn = an−2 ·bn−an ·bn+2.

That condition can be satisfied by choosing the two sequences (an)n∈N, (bn)n∈N asbeing linear in the indeterminate n. We choose

an :=µ +n

2, bn :=

µ−n2

with an arbitrary µ ∈ C\Z.


ii) The choice of µ assures that V contains no proper L-submodule, i.e. that Vis irreducible: Assume the existence of a non-zero submodule W ⊂V and choose anon-zero vector w∈W . The vector w is a linear combination of finitely many weightvectors of V . Hence w is contained in a finite direct sum V ′ of weight spaces. As aconsequence

w ∈W ∩V ′,

and W ∩V ′ is a non-zero, h-stable, and finite dimensional vector space. Hence theendomorphism induced by the action of h has an eigenvector. In particular, W ∩V ′

and a posteriori W contains a weight vector of V . Then W =V due to the definitionof V .

iii) Apparently, V has no maximal weight.

The Poincare-Birkhoff-Witt Theorem 8.8 allows to construct irreducible L-moduleswith arbitrary linear functionals

λ : H −→ C

as maximal weight. In general the resulting L-modules have infinite dimension.Therefore the next Section 8.3 will state necessary and sufficient conditions for finitedimensionality.

Theorem 8.21 (Existence of irreducible modules). Consider a semisimple Lie al-gebra L. For any linear functional λ ∈ H∗ exists a unique irreducible L-module Vwith a primitive element of weight λ .

Note that λ is not required to satisfy any condition with respect to integrality orpositivity. The proof of the theorem relies heavily on the existence and the propertiesof the universal enveloping algebra of L.

Proof. We first define a representation of the Borel-algebra B ⊂ L: We choosea 1-dimensional vector space

V λ := C · eB

and define

h.eB := λ (h) · eB for all h ∈ H and x.eB := 0 for all x ∈ N+.

Secondly, according to the universal property of UB the representation of B can bealso considered a representation of UB. By ring extension we define an UL-module

Vλ :=UL⊗UB V λ .

Here UL acts on the first component of the elementary tensors. Then Vλ is generatedby the element


e := 1⊗ eB ∈Vλ .

The latter element is not-zero because UL is a free UB-module having a basis whichextends the unit 1∈UL, see Proposition 8.9, part 4). The UL-module Vλ , sometimesnamed Verma module, can be considered also as an L-module.

In general the L-module Vλ is not irreducible. Therefore, the final step will factorout the proper L-submodules of Vλ : Consider a proper submodule

V ′ (Vλ .

Being an L-module, V ′ is in particular stable under H, hence we have the weightspace decomposition

V ′ = ∑µ weight of V ′

(V ′)µ ⊂ ∑µ weight of Vλ

(Vλ )µ

We claim that λ is not a weight of V ′. Otherwise e ∈V ′ which implies

V ′ =Vλ ,

a contradiction to V ′ being a proper submodule of Vλ . Therefore

V ′ ⊂ ∑µ weight of Vλ

µ 6=λ

(Vλ )µ

Define Nλ as the sum of all proper submodules of Vλ . Because e /∈Nλ the submodule Nλ (Vλ

is proper, and the quotientV :=Vλ/Nλ

is an irreducible L-module with heighest weight λ , q.e.d.

8.3 Finite-dimensional irreducible representations

We now investigate irreducible L-modules V with respect to finite dimensionality.Finite dimensionality can be characterized by the integrality of the weights of V :

• On one hand, integrality of the weights is a necessary condition for finite dimen-sionality due to Theorem 8.22.

• And on the other hand, Theorem 8.23 implies: If V has an integral and non-negative highest weight, then V is finite-dimensional.

We continue using the notation which has been fixed in the beginning of Section 8.2.All Lie algebras are assumed finite-dimensional.

8.3 Finite-dimensional irreducible representations 247

Theorem 8.22 (Finite-dimensional module: Integrality). Consider a complex semisim-ple Lie algebra L and a finite-dimensional L-module V . Then

1. The L-module V has the weight space decomposition

V =⊕

µ weight

V µ ,

and any weight µ ∈ H∗ of V is integral.

2. If V 6= 0 then V has a primitive element.

3. If V is standard cyclic then V is irreducible.

Proof. 1. Lemma 8.15 implies the direct sum decomposition of V .

For an arbitrary but fixed positive root α ∈Φ+ the vector space V is a finite-dimensional sl(2,C)-modulewith respect to

sl(2,C)' spanC < hα ,xα ,yα >⊂ L.

According to Theorem 5.8 the numbers µ(hα) are integers.

2. According to Theorem 3.20 the action of the Borel subalgebra B⊂ L on the finite-dimensional vector space V has a common eigenvector. The latter is a primitiveelement due to Lemma 8.17.

3. According to Proposition 8.18 the standard cyclic L-module V does not split asthe direct sum of distinct L-modules. Thanks to Weyl’s theorem on completereducibility, Theorem 4.24, the module V is irreducible, q.e.d.

Theorem 8.23 (Finite-dimensional module: Positivity). Consider a complex semisim-ple Lie algebra L and an irreducible L-module V with highest weight λ . Then V isfinite-dimensional iff for all α ∈Φ+

λ (hα) ∈ Z and λ (hα)≥ 0.

Proof. 1) Assume that V is finite-dimensional. For any positive root α ∈ Φ+ weconsider V as a sl(2,C)-module with respect to the Lie algebra

sl(2,C)' spanC < hα ,xα ,yα >⊂ L.

According to Corollary 5.7 the value λ (hα) is a non-negative integer.

2) Assume V 6= 0 and λ integral and non-negative. According to Theorem 8.22 aprimitive element e ∈V exists. Corollary 8.19 implies: All weights µ of V satisfy

µ(hα) ∈ Z, α ∈Φ+


because the Cartan integers are

hαi(α j) =< αi,α j >∈ Z.

i) The L-module V as sum of sl(2,C)-modules: We choose an arbitrary but fixedroot α ∈Φ+ and set

Sα := spanC < xα ,yα ,hα >' sl(2,C).

Setmα := λ (hα).

Ba assumption mα ∈ Z and mα ≥ 0. Consider the Sα -submodule of V generated bythe operation of Sα on a primitive element e ∈V λ of the L-module V , and inparticular the distinguished element

eα := ymα+1α .e ∈V.

On one hand, for any β ∈ ∆ ,β 6= α, the commutator satisfies

[xα ,yβ ] = 0

which impliesxβ .eα = xβ .(y

mα+1α .e) = ymα+1

α .(xβ .e) = 0,

because e is a primitive element. On the other hand, the formulas fromProposition 5.6 concerning the generated sl(2,C)-module show

xα .eα = 0.

In case eα 6= 0, also eα is a primitive element of V . It has the weight

λ − (mα +1)α.

But according to Corollary 8.19 it must have the same weight λ as the primitiveelement e. Therefore eα = 0. Then the vector space

spanC < yαi.e : 0≤ i≤ mα >⊂V

is a finite-dimensional Sα -module.

We denote by Tα the set of finite-dimensional Sα -submodules contained in V andby V (α) their sum. We just proved that Tα is not empty. We claim that V (α) isstable under the operation of L:If F ∈ Tα is a finite-dimensional vector space then also L.F is finite-dimensional. Itremains to show that L.F is an Sα -module, i.e.

Sα .(L.F)⊂ L.F,


or for any zα ∈ Sα and z ∈ L holds

zα .(z.F)⊂ L.F

Considering zα and z as endomorphisms of V we have

zα .z = z.zα +[zα ,z]

hencezα .(z.F)⊂ z.(zα .F)+ [zα ,z].F ⊂ L.F

because zα .F ⊂ F by assumption. As a consequence, also L.F is afinite-dimensional Sα -module, i.e.

L.F ∈ Tα .

Therefore V (α)⊂V is a non-zero L-module. The irreducibility of the L-module Vimplies

V (α) =V,

i.e. V is a sum of finite-dimensional Sα -modules.

ii) The Weyl group acts on the weight set: We show that the Weyl group permutesthe weights of V . Consider a weight µ and a non-zero element v ∈V µ . Accordingto part i) for any positive root α ∈Φ+ exists a finite-dimensional Sα -module Vα

with v ∈Vα . Consider the integer

pα := µ(hα) ∈ Z,

and set

x :=

(yα)

pα .v if pα ≥ 0

(xα)−pα .v if pα ≤ 0

According to Proposition 5.6 also x 6= 0 and x ∈Vα . The element x ∈V has theweight

µ− pα ·α = µ−< µ,α > α = σα(µ).

Hence also the functional σα(µ), the image under the Weyl reflection σα , is aweight of V . The claim follows because the Weyl group is generated by all Weylreflections.

iii) Finite weight set: To show that the weight set of V is finite consider an arbitrarybut fixed weight µ . According to Corollary 8.19 it has the form

µ = λ − ∑α∈∆

pα ·α, pα ∈ N for all α ∈ ∆ .


We shall show that the coefficients pα ,α ∈ ∆ , are bounded; hence there are onlyfinitely many.

Also the set−∆ := −α : α ∈ ∆

is a base of the root set Φ . According to Proposition 6.16 the Weyl group W actstransitively on the bases of Φ . Hence a Weyl reflection w ∈W exists with

w(∆) =−∆ .

According to part ii) also w(µ) is a weight of V . Using again Corollary 8.19 theweight has the form

w(µ) = λ − ∑α∈∆

qα ·α,qα ∈ N for all α ∈ ∆ .

Thenµ = w−1(w(µ)) = w−1(λ )− ∑

α∈∆

qα ·w−1(α) =

= w−1(λ )+ ∑α∈∆

rα ·α,rα ∈ N for all α ∈ ∆ .

The latter equality with the change of the sign is due to

w−1(∆) =−∆ .

Note thatqα : α ∈ ∆= rα : α ∈ ∆,

but for α ∈ ∆ possibly qα 6= rα because not necessarily w(α) =−α . We obtain

λ −w−1(λ ) =

(µ + ∑

α∈∆

pα ·α

)−

(µ− ∑

α∈∆

rα ·α

)= ∑

α∈∆

(pα + rα) ·α.

The linear functional λ −w−1(λ ) ∈ H∗ is independent from the choice of theweight µ . It has the basis representation

λ −w−1(λ ) = ∑α∈∆

cα ·α

with coefficientscα := pα + rα with pα ,rα ∈ N.

We obtain0≤ pα = cα − rα ≤ cα .

The estimate shows: All coefficients (pα)α∈∆ of the weight µ are bounded by theconstant

maxcα : α ∈ ∆


which is independent from µ . Hence there exist only finitely many weights.

iv) Weight space decomposition: The module V decomposes into finitely manyweight spaces according to part iii). Each weight space is finite-dimensional due toProposition 8.18. Hence V is finite-dimensional, q.e.d.

Definition 8.24 (Fundamental weights, dominant weights, and fundamental rep-resentations). Consider a complex semisimple Lie algebra L. With respect to thebase ∆ of its roots denote by

(hα)α∈∆

the corresponding basis of H.

1. The elements of the dual basis of H∗

(λα)α∈∆

are the fundamental weights of L with respect to (H,∆). The uniquely de-termined irreducible L-modules Vα with highest weight λα , α ∈ ∆ , are thefundamental L-modules or fundamental representations of L.

2. The dominant weights are the linear combinations of the fundamental weightswith non-negative integer coefficents

λ = ∑α∈∆

nα ·λα

with nα ∈ Z,nα ≥ 0, for allα ∈ ∆ .

Proposition 8.25 (Fundamental weights and Cartan matrix). Consider a semisim-ple complex Lie algebra L of rank = r and the base

∆ = α1, ...,αr

of its roots. Then the Cartan matrix of L relates the elements of ∆ and the funda-mental weights according to

(α1, ...,αr)> =Cartan(∆) · (λ1, ...,λr)

>.

Proof. In order to prove the claim one has to show for i = 1, ...,r, the equality oflinear functionals on H

αi =r

∑j=1

< αi,α j > ·λ j.

Evaluating both sides on the basis elements hk ∈ H,k = 1, ...,r, shows


• Left-hand side: According to Proposition 7.4, part 4)

αi(hk) =< αi,αk >

• Right-hand side:(r

∑j=1

< αi,α j > ·λ j

)(hk) =< αi,αk >, q.e.d.

Because all entries of the Cartan matrix of a semisimple Lie algebra L are inte-gers, Proposition 8.25 shows: The r-dimensional root lattice Λroot of L is a sublatticeof the lattice Λ with basis the fundamental weights.

The dominant weights are the highest weights of the finite-dimensional, irre-ducible L-modules. Hence the dominant weights characterize the finite-dimensionalirreducible L-modules. The result is a direct consequence of Theorem 8.23.

Theorem 8.26 (Classification of finite-dimensional irreducible modules). Con-sider a complex semisimple Lie algebra L of rank = r. Denote by (λα)α∈∆ the fun-damental weights of L with respect to (H,∆).

1. An irreducible L-module V with heighest weight λ is finite-dimensional iff λ isintegral and non-negative.

2. Mapping a finite-dimensional irreducible L-module V to its highest weight λ

defines a bijective map from the set of finite-dimensional irreducible L-modulesand the zero-module to the free Abelian monoid

Λ+ :=

r

∑α∈∆

nα ·λα ∈Λ : nα ∈ N for all α ∈ ∆

with basis the set of fundamental weights.

3. All weights of an irreducible finite-dimensional L-module have the form

µ = ∑α∈∆

nα ·λα ∈Λ ,nα ∈ Z for all α ∈ ∆ .

Proof. 1. See Theorem 8.23.

2. According to Theorem 8.21 for any linear functional λ ∈H∗ an irreducible L-modulewith highest weight λ exists. The weight λ has a basis representation

λ = ∑α∈∆

nα ·λα , nα ∈ C.


The L-module has finite dimension if and only if for all α ∈ ∆

λ (hα) = nα ∈ N

due to part 1).

3. According to Theorem 8.22 all weights µ of a finite-dimensional irreducible-module are integral, i.e. satisfy for all α ∈ ∆

µ(hα) ∈ Z, q.e.d.

Henceµ = ∑

α∈∆

µ(hα) ·λα , q.e.d.

Generalizing the notation for irreducible sl(2,C)-modules from Theorem 5.8 weintroduce the notation:

Notation 8.27 (Parametrization of irreducible L-modules). Consider a complexsemisimple Lie algebra L of rank = r. Denote by (λ j) j=1,...,r the fundamentalweights of L with respect to (H,∆). The irreducible finite-dimensional L-moduleof highest weight

λ =r

∑i=1

ni ·λi

is denotedV (n1, ...,nr).

Example 8.28 (Fundamental weights of sl(r+1,C)). Set L = sl(r+1,C).

i) According to Proposition 7.7 we may choose for L

• the Cartan subalgebra of traceless diagonal matrices

H := h ∈ d(r+1,C) : tr h = 0,

• and the base of the roots∆ = α1, ...,αr

withαi := (εi− εi+1)|H, i = 1, ...r,

and (ε j) j=1,...,r+1 the dual basis of the basis (E j, j) j=1,...,r+1 of d(r+1,C).

Then we have


• the positive roots

Φ+ = εi− ε j =

j

∑k=i

αi : 1≤ i < j ≤ r+1

with distinguished elements for α ∈Φ+

xα = Ei, j,yα = E j,i,hα = Ei,i−E j, j.

• the basis of H(hi := hαi := Ei,i−Ei+1,i+1)i=1,...,r

• and its dual basis (λi)i=1,...,r of fundamental weights

λi = ε1 + ...+ εi, i = 1, ...,r.

ii) In the specific case r = 2, i.e. L = sl(3,C), Lemma ?? implies

α1 = 2λ1−λ2, α2 =−λ1 +2λ2

orλ1 = (1/3)(2α1 +α2), λ2 = (1/3)(α1 +2α2),

see Figure 8.1.


Fig. 8.1 Lattice of fundamental weights of sl(3,C), monoid of dominant weights shaded

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47. Spanier, Edward: Algebraic Topology. Tata McGraw-Hill, New Delhi (Repr. 1976)48. Varadarajan, Veeravalli S.: Lie Groups, Lie Algebras, and their Representations. Springer,

New York (1984)49. Weibel, Charles A.: An Introduction to Homological Algebra. Cambridge University Press,

Cambridge (1994)

Index

1-parameter group, 41

Abelian, 37ad-nilpotent, 65ad-semisimple, 67algebraic multiplicity, 14

Banach algebra, 8Borel subalgebra, 237

Cartan criterionsemisimplicity, 100solvability, 94

Cartan decomposition, 147Cartan integer, 151Cartan matrix, 164Cartan subalgebra, 147Casimir element, 110center, 36centralizer, 36commutator algebra, 37Coxeter graph, 167

derivation, 40derivation

inner, 40outer, 40

derived algebra, 37derived series, 75descending central series, 71direct sum decomposition, 101dominant weight, 251dynamical Lie algebra of quantum mechanics,

86

eigenspace, 12exponential, 24

filtration, 222flag, 13fundamental weight, 251

generalized eigenspace, 12geometric series, 26graded algebra of the universal enveloping

algebra, 223

Heisenberg picture, 88highest weight, 240, 242

ideal, 36infinitesimal generator, 41integral weight, 238irreducible, 106

Jacobi identity, 36Jordan decomposition

abstract, 120Jordan decomposition

concrete, 15of adjoint representation, 66

Killing form, 92

L-module, 38Lie algebra

definition, 35nilpotent, 71of skew-Hermitian matrices, 43of skew-symmetric matrices, 42, 43of traceless matrices, 42semisimple, 96simple, 96solvable, 75

logarithm, 26

matrix

261

262 Index

diagonal, 37strictly upper triangular, 37upper triangular, 13, 37

matrix algebra, 37matrix group, 41minimial polynomial, 16

nilpotent, 13non-degenerate symmetric bilinear form, 99non-negative weight, 238normalizer, 36nullspace, 99

operator norm, 6orthogonal space, 99

Pauli matrix, 48Planck’s constant, 85Poincare-Birkhoff-Witt theorem, 232positive root, 147primitive element, 133, 239

radical, 76real form, 47reducible, 106representation

adjoint, 39definition, 38faithful, 38

root, 141, 151root

simple, 155root space, 141root space decomposition, 141root system

base, 155definition, 150

Weyl group, 152root vector, 141

semidirect product, 82semisimple, 13set of roots, 141simply connected, 50solvable ideal, 75special linear group, 42special orthogonal group, 42, 43special unitary group, 43standard cyclic module, 239subalgebra, 36submodule, 106symmetric algebra, 221symplectic group, 42

tensor algebra, 220Theorem of

Engel, 72Lie, 81Weyl, 113

toral subalgebradefinition, 128maximal, 128

trace formdefinition, 92

universal enveloping algebra, 220

Verma module, 246

weight, 133weight space, 133weight vector, 133, 238Weyl group, 152

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