lie groups and algebras i lecture notes for mth 91...

Lie Groups and Algebras ILecture Notes for MTH 91

F13

Ulrich Meierfrankenfeld

January 15, 2014

Contents

1 Basic Properties of Lie Algebras 51.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Structure constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3 Derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.4 Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.5 The universal enveloping algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.6 Nilpotent Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.7 Finite Dimensional Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2 The Structure Of Standard Lie Algebras 372.1 Solvable Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.2 Tensor products and invariant maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.3 A first look at weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.4 Minimal non-solvable Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.5 The simple modules for slpK2q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.6 Non-degenerate Bilinear Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512.7 The Killing Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542.8 Non-split Extensions of Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592.9 Casimir Elements and Weyl’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612.10 Cartan Subalgebras and Cartan Decomposition . . . . . . . . . . . . . . . . . . . . . . . . 622.11 Perfect semsisimple standard Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3 Rootsystems 713.1 Definition and Rank 2 Rootsystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713.2 A base for root systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833.3 Elementary Properties of Base . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 853.4 Weyl Chambers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4 Uniqueness and Existence 934.1 Simplicity of semisimple Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 934.2 Generators and relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

3

4 CONTENTS

Chapter 1

Basic Properties of Lie Algebras

1.1 DefinitionDefinition 1.1.1. Let pR,`q be an abelian group and

r , s : Rˆ R Ñ R, pa, bq Ñ ra, bs

a binary operation on R. Then we say that the Jacobi Identity holds for pR,`, r , sq if

ra, rb, css ` rb, rc, ass ` rc, ra, bss “ 0,

for all a, b, c P R.

Example 1.1.2. Let R be a ring and for a, b P R define

ra, bs :“ ab´ ba.

( ra, bs is called the commutator or bracket of a and b.). Then the Jacobi Identity holds for pR,`r , sq. Indeed,

ra, rb, css ` rb, rc, ass ` rc, ra, bss

“ apbc´ cbq ´ pbc´ cbqa` bpca´ acq ´ pca´ acqb` cpab´ baq ´ pab´ baqc

“ abc´ acb´ bca` cba` bca´ bac´ cab` acb` cab´ cba´ abc` bac

“ 0

Definition 1.1.3. Let K be a field, A a (left) vector space over K and ¨ : A ˆ A Ñ A a K-bilinear function.Then pA, ¨q is called a K-algebra. If ¨ is associative, then A is called an associative algebra.

Definition 1.1.4. Let V and W be vector spaces over the field K and

s : V ˆ V Ñ W, px, yq Ñ spx, yq

a bilinear function.

(a) s is called symplectic, if spx, xq “ 0 for all x P V.

(b) s is called alternating, if spx, yq “ ´spy, xq for all x, y P V.

5

6 CHAPTER 1. BASIC PROPERTIES OF LIE ALGEBRAS

Lemma 1.1.5. Let V and W be vector spaces over the fieldK and s : VˆV Ñ W a bilinear function.

(a) If s is symplectic, then s is alternating.

(b) If charK ‰ 2, then s is symplectic if and only if s is alternating.

Proof. (a) Suppose s is symplectic. Then for all x, y P V:

0 “ spx` y, x` yq “ spx, xq ` spx, yq ` spy, xq ` spy, yq “ spx, yq ` spy, xqq

and so spx, yq “ ´spx, xq.

(b) Suppose that s is alternating. Then spx, yq “ ´spy, xq and using y “ x gives spx, xq “ ´spx, xq andso 2spx, xq. If charK ‰ 2 this implies spx, xq “ 0 and s is alternating. �

Definition 1.1.6. A K-algebra pA, r , sq is called a Lie algebra provided that r , s is symplectic and r , s fulfillsthe Jacob identity.

Example 1.1.7. Let K be any field.

(1) Let A be an associative K-algebra and define ra, bs “ ab ´ ba. Then pA, r , sq is a Lie-algebra. Wedenote that the Lie Algebra by LpAq.

(2) In the special case A “ EndKpVq, we denote LpAq be glKpVq.

(3) A Lie algebra L is called Abelian if ra, bs “ 0 for all a, b P L. Note that K-space V becomes an abelianLie algebra if one defines ra, bs “ 0 for all a, b P V .

Definition 1.1.8. Let K be a field, let A and A1 be K-algebras and B Ď A.

(a) A K-algebra homomorphism f : A Ñ A1 is a K-linear function with f pabq “ f paq f pbq for all a, b P A.

(b) B is called a K-subalgebra of A if B is a K-subspace and bc P B for all b, c P B.

(c) B is called left (right) K-ideal in A if B is a K-subspace of B and ab P B ba P Aq for all a P A andb P B. B is a K-ideal in A if B is a left and right K-ideal in A.

Example 1.1.9. Let V be a K-space.

(a) Let W be K-subspace of V and define

A :“ t f P EndKpVq | f pWq Ď Wu

B :“ t f P EndKpVq | W ď ker f u

C :“ t f P EndKpVq | Im f ď Wu

D :“ BXC

Then A is a K-subalgebra of EndKpVq and B and C are ideals of A with respect to ˝ and with respect tor , s. pD, ˝q is a commutative K-algebra and pD, r , sq is an abelian Lie-algebra.

(b) LetW be a set of K-subspaces of V . Then

EndKpV,Wq :“ t f P EndpVq | f pWq ď W for all W PWu

is a K subalgebra of EndKpVqqWe denote the corresponding Lie algebra by glpVq by glpV,Wq.

1.2. STRUCTURE CONSTANTS 7

(c) Suppose that V has a finite basis pv1, v2, . . . , vnq andW consist of the n`1 subspaceKv1`Kv2`...`Kvi,0 ď i ď n. Then glpV,Wq now consist of all the upper triangular matrices (with respect to the givenbasis).

(d) Let f be a bilinear form on V , that is a K-bilinear function f : V ˆ V Ñ K. Define

clp f q “ tα P glpVq | f pαv,wq ` f pv, αwq “ 0 for all v,w P Vu.

We claim that clp f q is a Lie subalgebra of glpVq. Clearly it’s a K-subspace. Let α, β P clp f q andv,w P V . Then

f prα, βsv,wq “ f pαβv,wq ´ f pβα,wq

“ ´ f pβv, αwq ` f pαv, βwq

“ f pv, βαwq ´ f pv, αβwq

“ ´ f pv, rα, βswq

So rα, βs P clp f q and clp f q is indeed Lie subalgebra of glKpVq.

1.2 Structure constantsDefinition 1.2.1. Let I be a set.

(a) Let pKiqiPI be a family of sets. ThenŚ

iPI Ki is the set of all functions k with domain I and kpiq P Ki forall i P I. We usually denote such a k as pkiqiPI where ki “ kpiq.

Ś

iPI Ki is called the direct product ofpKiqiPI .

(b) Let pKiqiPI be a family of groups.À

iPI Ki is the set of all functions k with domain I such that kpiq P Ki

for all i P I and ti | kpiq ‰ 1Kiu is finite.À

iPI Ki is called the direct sum of pKiqiPI .

(c) Let K be a set. Then K I denotes the set of all functions from I to K. So K I “Ś

iPI KI .

(d) Let K be group, then KI denotes the set of all functions from I to K such that ti P I | f piq ‰ 1Ku isfinite. So KI “

À

iPI Ki.

Definition 1.2.2. pL, r , sq be a K-algebra and B a basis for L. For i, j P B let paki jqkPB be the coordinate

vector of ri, js with respect to the basis B. So aki j P B, ak

i j “ 0 for all but finitely many k P B and

ri, js “ÿ

kPB

aki jk.

Lemma 1.2.3. Let pL, r , sq be a K-algebra, B a basis for L and aki j, i, j, k P B. the structure constants of L

with respect to B. Then pL, r , sq is a Lie-algebra if and only if the following three statements hold:

(i) akii “ 0 for all i, k P B,

(ii) aki j ` ak

ji “ 0 for all i, j, k P B, and

(iii)ř

mPB ami ja

lkm ` am

jkalim ` am

kialjm “ 0 for all i, j, k, l P B.


Proof. Since r , s is K-bilinear, pL, r , sq is a Lie-algebra if and only if

(I) ri, is “ 0 for all i P B.

(II) ri, js ` r j, is “ 0 for all i, j P B, and

(III) rk, ri, jss ` ri, r j, kss ` r j, rk, iss “ 0 for all i, j, k P B.

Indeed the first two statement hold if and only if r , s is symplectic and the third one if and only if theJacobi identity holds.

Since ri, is “ř

kPB akiik, (I) is equivalent to (i).

Since ri, js ` r j, is “ř

kPB

´

aki j ` ak

ji

¯

k, (II) is equivalent to (ii).

Also

rk, ri, jss “ rk,ÿ

mPB

ami jms “

ÿ

lPB

˜

ÿ

mPB

ami ja

lkm

¸

l

Formulas for ri, r j, kss and r j, rk, iss are obtained by cyclic permutation of pi, j, kq. Thus (III) is equivalentto (iii). �

Definition 1.2.4. (a) Let V be a K-space and W Ď V. Then xWyK denotes the K-subspace of V spannedby W, and xWy denotes the additive subgroup of V generated by V.

(b) Let L be a Lie-algebra over K and A, B Ď L. Then

rA, Bs :“

ra, bsˇ

ˇ a P A, b P B(

Lemma 1.2.5. Let L be a Lie-algebra over K and A, B Ď L.

(a) xrA, BsyK “ xrxAyK, xByKsy.

(b) Suppose L “ xAyKy and that ď is a total ordering on A. Then

xrL, Lsy “ xra, bs | a ă b P AyK

Proof. (a) This holds since r , s is K-bilinear.(b) Since r , s is symplectic and so alternating, ra, as “ 0 and rb, as “ ´ra, bs for all a, b P A. Thus

xra, bs | a ă b P AyK “ xra, bs | a, b P AyK “ xrA, Asy

and so (b) follows from (a). �

Lemma 1.2.6. (a) Let L be 2-dimensional non-abelian Lie-algebra over K. Then L has a basis pa, bq withra, bs “ a.

(b) Up to isomorphism there exists a unique 2-dimensional non-abelian K-algebra.

1.3. DERIVATIONS 9

Proof. (a) Let px, yq be a basis for L and put a :“ rx, ys. By 1.2.5(b)

xrL, Lsy “ xayK.

So if a “ 0 then L is abelian contrary to the assumptions. So a ‰ 0 and we can extended a to basis pa, bqof L. ra, bs “ ka for some 0 ‰ k P K. Replacing b by k´1b we may assume ra, bs “ a and so (a) holds.

(b) : (a) show that up isomorphism there exists at most one 2-dimensional non-abelian Lie Algebra.To show that existence we will exhibit such Lie -algebra as a Lie-subalgebra of glpK2q. Namely choose

a :“

»

–

0 0

1 0

fi

fl b :“

»

–

0 0

0 ´1

fi

fl .

Then ra, bs “ a. Put L “ xa, byK. Then rL, Ls Ď xayK Ď L and so L a Lie-subalgebra of glpK2q. Itfollows that L is a non-abelian 2-dimensional Lie-algebra and so (b) holds. �

1.3 DerivationsDefinition 1.3.1. Let A be a K-algebra. Then a derivation of A is a K-linear function δ : A Ñ A with

δpabq “ δpaqb` aδpbq

for all a, b P A. derpAq denotes the set of all derivations of A.

Lemma 1.3.2. Let A be a K-algebra. Then derpAq is a K-subalgebra of glpAq.

Obviously derpAq is a K-subspace of glpAq. Now let γ, δ P derpAq and a, b P A. Then

rγ, δspabq “ γδpabq ´ δγpabq

“ γpaδpbqq ` γpδpaqbq ´ δpaγpbqq ´ δpγpaqbq

“ γpaqδpbq ` aγpδpbqq ` γpδpaqqb` δpaqγpbq

´δpaqγpbq ´ aδpγpbqq ´ δpγpaqqb´ γpaqδpbq

“ apγpδpbqq ´ δpγpbqqq ` pγpδpaqq ´ δpγpαqqqb

“ rγ, δspaqb` arγ, δspbq

Lemma 1.3.3. Let A be an assocative K-algebra and for a P A define

lpaq : A Ñ A, b Ñ ab,

rpaq : A Ñ A, b Ñ ba,

ad paq : A Ñ A, b Ñ ra, bs

Let a, b, c P A. Then

(a) ad paq “ lpaq ´ rpaq.

(b) lpaq, rpaq and ad paq all are K-linear.

(c) ra, bcs “ ra, bsc` bra, cs, that is, ad paq is a derivation of A


(d) l : A Ñ EndpAq, a Ñ lpaq is a homomorphism.

(e) r : A Ñ EndpAq, a Ñ rpaq is an anti-homomorphism.

Proof. (a) and (b) are readily verified.

(c) We compute

bra, cs ` ra, bsc “ bac´ bca` abc´ bac “ apbcq ´ pbcqa “ ra, bcs.

(d) and (e) are readily verifed as A is associative. �

Lemma 1.3.4. Let a P L. Define ad paq : L Ñ L, l Ñ ra, ls

(a) ad paq is a derivation of L.

(b) Let δ P derpLq. Then rδ, ad paqs “ ad pδpaqq.

(c) ad : L Ñ glpLq, l Ñ ad plq is a K-algebra homomorphism.

(d) ad pLq is an ideal in derpLq.

Proof. Let a, b, c P A. Then

ad paqprb, csq “ ra, rb, css “ ´rb, rc, ass ´ rc, ra, bss “ rb, ra, css ` rra, bs, cs “ rb, ad apcqs ` rad paqpbq, cs.

and so ad paq is a derivation.Let δ be a derivation of A. Then

rδ, ad paqspbq “ δpad paqpbqq ´ ad paqpδpbqq“ δpra, bsq ´ ra, δpbqs

“ rδpaq, bs ` ra, δpbqs ´ ra, δpbqs

“ ad pδpaqqpbq

Thus (b) holds.From (b) applied to the derivation ad b in place of δ we have

“

ad pbq, ad paq‰

“ ad`

rb, as˘

so (c) holds.Finally (d) follows from (b). �

A derivation of the form ad paq is called an inner derivation. All other derivations of a Lie Algebra arecalled outer derivations.

1.4 ModulesDefinition 1.4.1. Let A be an associative K-algebra and V a K-space.

(a) A K-algebra representation for A on V is a K-algebra homomorphism Φ : A Ñ EndKpVq.

(b) A K-algebra action for A on V is a K-bilinear function

Aˆ V Ñ V, pa, vq Ñ av

such thatpabqv “ apbvq

for all a, b P A, v P V.

1.4. MODULES 11

(c) If ¨ is an K-algebra action of A in V, then pV, ¨q is called a K-algebra module for A. Often we will beless precise and just say that V is an A-module.

Definition 1.4.2. Let V be a K-space.

(a) A K-Lie-algebra representation for L on V is a K-algebra homomorphism α : L Ñ glpVq.

(b) A K-Lie algebra action for L on V is a K- bilinear function

Lˆ V Ñ V, pa, vq Ñ av

such thatra, bsv “ apbvq ´ bpavq

for all a, b P L, v P V.

(c) If ¨ is an K-Lie-algebra action of L in V, then pV, ¨q is called a K-Lie algebra module for A. Often wewill be less precise and just say that V is an L-module.

Note that Lie algebra L can be associative (namely if rL, rL, Lss “ 0). In this case a K-algebra module forL does not have to be K-Lie Algebra-module for L (and vice versa).

Example 1.4.3. (1) Let V be vector space over K. Then V is a algebra module for EndKpVq and a Lie-algebra module for glKpVq via f v “ f pvq for all f P EndKpVq and v P V .

(2) Let A is a associative algebra, then by 1.3.3 the left multiplication l is a representaion for A on A.

(3) If L is a Lie algebra then by 1.3.4 ad is a representation of L on L.

Lemma 1.4.4. Let A be an associative or a Lie algebra and V a K-space.

(a) Let Φ be a representation for A on V. Then

Aˆ V Ñ V, pa, vq Ñ Φpaqpvq

is an action for A on V.

(b) Suppose Aˆ V Ñ V, pa, vq Ñ av is an action for A on V. For a P A define

Φpaq : V Ñ V, v Ñ av

then

Φ : A Ñ EndpVq, a Ñ Φpaq

is a representation for A over V.

Proof. Straightforward. �

Lemma 1.4.5. Let V be a module for the associative algebra A. Then with the same action V is also a modulefor LpAq. In particular, left multiplication is an action of the Lie-Algebra LpAq on A.


Proof. Let a, b P A and v in v. Then

ra, bsv “ pab´ baqv “ apbvq ´ bpavq.

�

Definition 1.4.6. Let V be a module for the associative or Lie algebra A. Let U,W Ď V and B Ď A.Then

(a) AnnWpBq :“ tw P W | bw “ 0 for all b P Bu and AnnWpB,Uq :“ tw P W | bw P U for all b P Bu

(b) AnnBpWq :“ tb P B | bw “ 0 for all w P Wu and AnnBpW,Uq :“ tb P B | bw P U for all w P Wu

(c) NBpWq :“ tb P B | bw P W for all w P Wu.

(d) BW :“ tbw | b P B,w P Wu.

(e) W is called B-invariant if BW Ď W.

(f) W is called an A-submodule of V, if W is an A-invariant K-subspace of V.

(g) If AnnApVq “ 0 we say that V is a faithful A-module.

It might be interesting to note that AnnWpBq “ AnnWpB, 0q, AnnBpWq “ AnnBpW, 0q and NLpWq “AnnLpW,Wq.

Definition 1.4.7. (a) Let L be a Lie-algebra. Then ZpLq “ ta P L | ra, bs “ 0 for all b P Lu.

(b) Let R be a ring. Then ZpRq “ ta P R | ab “ ba for all b P Ru.

(c) Let A be a K-algebra and B,D Ď A. D is called left B-invariant if BD Ď D, right B-invariant ifDB Ď D and B-invariant if DBY BD Ď D.

Note that for an associative K-algebra L, ZpAq “ ZpLpAqq.

Remark 1.4.8. Let L be a Lie-algebra and view L as L-module via the adjoint. Let B,D Ď L and let U be aK-subspace of L. action.

(a) ZpLq “ AnnLpLq.

(b) AnnBpD,Uq is unambiguous. It could be mean

tb P B | rb, ds P U for all d P Bu or tb P B | rd, bs P U for all d P Bu

But since rb, ds “ ´rd, bs and U is a subspace of L, these two sets are the same.

(c) If D “ ´D or B “ ´B, D is left B-invariant if and only of D is right B-invariant.

(d) D is left B-invariant if and only if D is an B-invariant subset of the adjoint module L.

Lemma 1.4.9. Let A be a associative or Lie-algebra over K and V a module for A. Let B,D Ď A, let U be aK-subspace of V and let U Ď W Ď V.

(a) W Ď AnnVpB,Uq if and only of B Ď AnnApW,Uq.

(b) Suppose U is D-invariant and B is right D-invariant. Then AnnVpB,Uq is D-invariant K-subspace ofV.

1.4. MODULES 13

(c) Suppose B is right ideal in A. Then AnnVpBq is an A-submodule of V.

(d) Let W˚ be the K-subspace of V generated by W. Then AnnBpW,Uq “ AnnBpW˚,Uq.

(e) Suppose U and W are D-invariant. Then AnnApW,Uq is a D-invariant K-subalgebra of A.

(f) NApUq is a K-subalgebra of V and AnnApUq and AnnApV,Uq are ideals in NApUq.

(g) Let B˚ be the K-subalgebra of L generated by A. Then AnnWpB,Uq “ AnnWpB˚,Uq.

Proof. We prove the lemma for the case that L :“ A is a Lie Algebra and leave the (easier) associativeAlgebra case to the reader.

(a) Both statement are equivalent to bw P U for all b P B,w P W.

(b) Let v P AnnVpB,Uq, b P B and d P D. By definition of AnnVpB,Uq, bv P U and since U isD-invariant, dpbvq P U. Also as B is right D invariant, rB,Ds Ď B and and so rb, dsv P U.

bpdvq “ dpbvq ` rb, dsv P U.

Thus dv P AnnVpB,Dq and AnnVpB,Dq is D-invariant. Since L acts K-linearly on V , AnnVpB,Uq is aK-subspace of V and so (b) holds.

(c) Just apply (b) with D “ A and U “ 0.

(d) Note that W Ď AnnVpAnnBpW,Uqq. By (b) the latter is a K-subspace of V . Hence

W˚ ď AnnVpAnnBpW,Uqq and so AnnBpW,Uq ď AnnBpW˚,Uq.

The reverse inclusion is obvious and so (d) holds.

(e) Let P AnnLpW,Uq, d P D and w P W. Since W is D-invariant, dw P W and so lpdwq P U. Sincelw P U and U is D-invariant, dplwq P U. Thus

rd, lsw “ dpbwq ´ bpdwq P U.

So rd, ls P AnnLpW,Uq and AnnLpW,Uq is D-invariant.

Let a, b P AnnLpW,Uq, k P K and w P W. Then aw P U and bw P U. Since U is a K-subspace,kpawq “ kpawq P U and pa ` bqw “ aw ` bw P U. So AnnLpW,Uq is K-subspace of L. Since U Ď

W, aU Ď aW Ď U Ď W. So both U and W are AnnLpW,Uq -invariant. It follows that AnnLpW,Uq isAnnLpW,Uq-invariant, that is

rAnnLpW,Uq,AnnLpW,Uqs ď AnnLpW,Uq.

and so AnnLpW,Uq is indeed a K-subalgebra of L.

(f) Since NLpUq “ AnnLpU,Uq, (e) shows that NLpUq is K-subalgebra of L. By (e) applied with D “

NLpUq, W “ V and U “ U, AnnLpV,Uq is NLpUq-invariant K-subalgebra of L. Note that AnnLpV,Uq ďNLpUq and so AnnLpV,Uq is an ideal in NLpUq. Similarly, by (e) applied with D “ NLpUq, W “ U andU “ 0, AnnLpUq is NLpUq-invariant K-subalgebra of L. Note that AnnLpUq ď NLpUq and so AnnLpUq is anideal in NLpUq.

(g) Note that B Ď AnnLpAnnWpB,Uqq. By (e) the latter is a K-subalgebra of L. Hence

B˚ ď AnnLpAnnWpB,Uqq and so AnnWpB,Uqq ď AnnWpB˚,Uq.

The reverse inclusion is obvious and so (g) holds. �


Lemma 1.4.10. Let V be a module for the associative or Lie algebra A.

(a) Let Φ is the representation corresponding to the module V, then AnnApVq “ kerΦ.

(b) V is a faithful A{AnnApVq module via

pa` AnnApVqqv “ av

for all a P AnnApVq and v P V.

(c) Let W be an A-submodule of V. Then V{W is an A-module via

apv`Wq “ av`W

for all a P A, v P V.

Proof. Readily verified. �

Lemma 1.4.11. Let A be a Lie or an associative algebra, V an A-module, B a subset of A and W Ď V withBW Ď xWyK. Then xWyK is B-invariant.

Proof. By 1.4.9(b) Z :“ AnnVpB, xWyKq is a K-subspace of V . Note that W Ď Z and so xWyK Ď Z. ThusBxă WyK Ď BZ Ď xWyK and xWyK is B-invariant. �

Lemma 1.4.12. Let A be a Lie or an associative algebra, V an A-module, and D, E Ď L. Let W be D-invariant K subspace of V. Suppose that D is left E-invariant and if A is a Lie-algebra, that W is E-invariant.Then for all n P N, xDnWy is an E Y D-invariant K-subspace of V.

Proof. By induction on n it suffices to treat the case n “ 1. Note that for each d P D, dW is K-subspace ofW and so also xDWy “ xdW | d P Dy is a K-subspace of V . In particular, xDWy “ xDWyK.

If A is associative, then

EpDWq “ pEDqW Ď DW Ď xDWy,

and if A is a Lie Algebra then

EpDWq Ď DpEWq ` rE,DsW Ď DW ` DW Ď xDWy

Thus 1.4.11 shows that xDWy is E-invariant. Since W is D-invariant, DW Ď W and so DpDWq Ď DW Ď

xDWy. So xDWy is also D-invariant. �

1.5 The universal enveloping algebraWe assume the reader to be familiar with the elementary properties tensor products and symmetric powers,see for example [La].

Definition 1.5.1. Let V, W be K-space, I a set and pViqiPI a family of K-spaces. Let J,K Ď I with I “ JY Kand J X K “ H.

(a) For J Ď I put VJ :“Ś

jPJ V j.

(b) For vJ “ pv jq jPJ P VJ and vK “ pvkqkPK P VK , denote pviqiPI P VI by vJ Y vK .

1.5. THE UNIVERSAL ENVELOPING ALGEBRA 15

(c) f : VI Ñ W be a function and b P VK . Then fb denotes the function

fb : VJ Ñ W, a Ñ f paY bq

(d) A function f : VI Ñ W be is called K-multilinear if for all i P I and all b P VIztiu, fb : Vi Ñ W isK-linear.

(e) A function f : V I Ñ W is called symmetric, if

f pvq “ f pv ˝ πq

for all v “ pviqiPI P V I and π P SympIq.

Definition 1.5.2. Let pViqiPI be a family of K-spaces, W a K-space and

f :ą

iPI

Vi Ñ W

a K-multilinear function. Then pW, f q is a called a tensor product of pViqiPI over K provided that for allK-spaces W 1 and all K-multilinear function

g :ą

iPI

Vi Ñ W 1,

there exists a unique K-linear function g : V Ñ W 1 with g “ g ˝ f .

Lemma 1.5.3. Let pViqiPI be a finite family of K-spaces.

(a) Up to isomorphism, there exists a unique tensor product pW, f q of pViqiPI over K. We denote W byÂ

iPI Vi and f by b. If v “ pviqiPI PŚ

iPI Vi we will write bv and biPIvi for f pvq. If I “ t1, . . . , . . . nuwe will also write v1 b v2 b . . .b vn for f pvq.

(b) For i P I let Bi be a (unordered) basis of Vi over K. Then

!

biPI bi

ˇ

ˇ

ˇpbiqiPI P

ą

iPI

Bi

)

is basis forÂ

iPI Vi over K.

Definition 1.5.4. Let V and W be K-spaces, I a set and

f : V I Ñ W

a symmetric K-multilinear function. Then pW, f q is a called an I-th symmetric power of V over K providedthat for all K-spaces W 1 and all symmetric K-multilinear function

g : V I Ñ W 1,

there exists a unique K-linear function g : V Ñ W 1 with g “ g ˝ f .

Lemma 1.5.5. Let V be a K-space and I a finite set.

(a) Up to isomorphism, there exists a unique I-th symmetric power pW, f q of V over K. We denote W byS IV. and f by

ś

. If v “ pviqiPI P V I we will writeś

v andś

iPI vi for f pvq. If I “ t1, . . . , . . . nu wewill also write v1v2 . . . vn for f pvq.


(b) Let B be an ordered basis for B. Then

!

b1b2 . . . bn

ˇ

ˇ

ˇb1 ď b2 ď . . . ď bn P B

)

is basis for S nV.

Definition 1.5.6. Let V be a K-space and T an associative K algebra with identity and

f : V Ñ T

a K-linear function. Then pT, f q is a called a tensor algebra for V over K provided for all associativeK-algebras with identity T 1 and all K-linear function

g : V Ñ T 1,

there exists a unique K-algebra homomorphism g : T Ñ T 1 with g “ g ˝ f .

Lemma 1.5.7. Let V be a K-space.

(a) Up to isomorphism, there exists a unique tensor algebra pT, f q of V over K. We denote T be T pVq. fis 1-1 and we identify v P V with f pvq in T and denote the multiplication in T by b. We also identifyk P K with k1T in T .

(b) Let B be basis for B, then

!

b1 b b2 b . . .b bn

ˇ

ˇ

ˇn P N, b1, b2 . . . bn P Bu

is basis for T pVq.

Proof. The uniqueness statements follows easily from the definitions.(a) Define T “

À8

n“0Ân V . Identify

Ân V with its image in T and V withÂ1 V . Define a multiplication

b on T by extending the K-bilinear function

b :mâ

V ˆ

nâ

Vm Ñ

m`nâ

V`

pv1 b . . .b vmq, pw1 b . . .wnq˘

Ñ v1 b . . .bvm b w1 b . . .b wn.

to a K-bilinear function T ˆ T Ñ T . Then its is staighforward to check that T is an associative algebra. Notethat

Â0 V is a 1-dimensional K-space with basis bH and bH is an identity in T .Suppose T 1 is an associative K- algebra with 1 and g : V Ñ T 1 is linear. Define g : T Ñ T 1 by

gpv1 b v2 b . . .b vmq “ gpv1qgpv2q . . . gpvmq

Again its straight forward to check that g is the unique K-algebra homomorphism from T to T 1 withg “ g ˝ idV . So pT, idVq is a tensor algebra for V over K.

(b): For given n P N, 1.5.3(b) shows that!

b1 b b2 b . . .b bn

ˇ

ˇ

ˇb1, b2 . . . bn P Bu

is a K-basis forÂn V . Since T “

À8

n“0Ân V , the union of these bases forms a basis for T . �


Definition 1.5.8. Let V be a K-space and S an commutative associative K algebra with identity and

f : V Ñ S

a K-linear function. Then pS , f q is a called a symmetric algebra for V over K provided for all commutative,associative K-algebras with identity S 1 and all K-linear function

g : V Ñ S 1,

there exists a unique K-algebra homomorphism g : S Ñ S 1 with g “ g ˝ f .

Lemma 1.5.9. Let V be a K-space.

(a) Up to isomorphism, there exists a unique symmetric algebra pS , f q of V over K. We denote S be S pVq.f is 1-1 and we identify v P V with f pvq in S and denote the multiplication in S by ¨. We also identifyk P K with k1S in S .

(b) Let B be an ordered basis for B, then

!

b1 b b2 b . . .b bn

ˇ

ˇ

ˇn P N, b1 ď b2 ď . . . ď bn P Bu

is basis for S pVq.

Proof. The uniqueness statements follows easily from the definitions.(a) Define S “

À8

n“0 S nV . Identify S nV with its image in S and V with S 1V . Define a multiplication ¨on S by extending the K-bilinear function

b : S mV ˆ S nVm Ñ

m`nâ

V`

pv1 . . . vmq, pw1s . . .wnq˘

Ñ v1 . . .vm b w1 . . .wn.

to a K-bilinear function S ˆ S Ñ S . Then its is staighforward to check that S is an commutative associativealgebra. Note that S 0V is a 1-dimensional K-space with basis

ś

H andś

H is an identity in T .Suppose S 1 is an commutative and associative K- algebra with 1 and g : V Ñ S 1 is K-linear. Define

g : S Ñ S 1 bygpv1v2 . . . vmq “ gpv1qgpv2q . . . gpvmq

Again its straight forward to check that g is the unique K-algebra homomorphism from T to T 1 withg “ g ˝ idV . So pS , idVq is a symmetric algebra for V over K.

(b): For given n P N, 1.5.3(b) shows that!

b1b2 . . . bn

ˇ

ˇ

ˇb1 ď b2 ď . . . ď bn P Bu

is a K-basis for S nV . Since S “À8

n“0 S nV , the union of these bases forms a basis for S . �

Definition 1.5.10. Let L be Lie-algebra and U an associative K algebra with identity and

f : L Ñ LpUq

a K-algebra homomorphism. Then pU, f q is a called a universal enveloping algebra for L over K providedfor all associative K-algebras with identity U 1 and all K-algebra homomorphism function

g : L Ñ LpU 1q

there exists a unique K-algebra homomorphism g : U Ñ U 1 with g “ g ˝ f .


Lemma 1.5.11.Let L be a Lie algebra over K. Then L has a universal enveloping algebra pU, f q and pU, f q is unique up toisomorphism.

Proof. Let T be the tensor algebra for L over K. Let I be the ideal in LpT q generated by all

ab b´ bb a´ ra, bs, a, b P L

Then T{I is a universal enveloping algebra. �

Lemma 1.5.12. Let pU, f q a universal enveloping algebra of the Lie algebra L over K.

(a) Let V an L-module. Then there exists a unique action of U on V with f plqv “ lv for all l P L

(b) Let V be an U-module. Then V is an L-module via lv “ f plqv for all l P V and v P V.

Proof. (a): Let g : L Ñ glpVq be the homomorphism associated to the L-module V . Since glpVq “LpEndKpVqq, the universal property of pU, f q shows that there exists a unique K-algebra homomorphism,g : U Ñ EndKpVq with g “ g ˝ f . Note that this just means that there exists a unique action of U on V withlv “ f plqv for all l P L, v P V .

(b): Since V is a module for U, V is also a module for LpUq (see 1.4.5). Since f is K-algebra homomor-phism and composition of homomorphism are homomorphism we conclude that pl, vq Ñ f plqv is indeed anaction of L on V . �

Lemma 1.5.13. Let pU, f q be an universal enveloping algebra for Lie-algebra L. Also let B be an orderedbasis for L. Put

Um “ x f pLqi | 0 ď i ď myK.

(a) U “Ť8

m“0 Um..

(b) Um “ x f pb1q . . . f pbiq | 0 ď i ď m, b1 ď b2 ď . . . ď bi P ByK.

(c) U “ x f pb1q . . . f pbmq | m P N, b1 ď b2 ď . . . ď bm P ByK.

Proof. (a) Put U 1 “Ť8

m“0 UM . Since f pLqi f pLq j “ f pLqi` j, U 1 is a subalgebra of U. Also f pLq Ď U 1 andso by definition of U there exists a K-algebra homomorphism f : U Ñ U 1 with f “ f ˝ f . Hence both f andidU are K-algebra homomorphism from U to U and f “ f ˝ f “ idU ˝ f . So the uniqueness assertion in thedefinition of pU, f q shows that f “ idU . Thus U “ Im idU “ Im f Ď U 1 and U “ U 1..

Put Vm “ x f pb1q . . . f pbiq | 0 ď i ď m, b1 ď b2 ď . . . ď bi P ByK and suppose inductively thatUm´1 “ Vm´1. Since B is a basis for L,

Um “ x f pb1q . . . f pbiq | 0 ď i ď m, b1, b2, . . . , bi P By.

and so it suffices to show that f pb1q . . . f pbmq P Vm for all b1, b2, . . . , bm P B. Put di “ f pbiq and let 0 ď i ă n.Then

rdi, di`1s “ r f pbiq, f pbi`1qs “ f prbi, bi`1sq P f pLq,

d1 . . . di´1rdi, di`1sdi`2 . . . dn P f pLqm´1 Ď Um´1 “ Vm´1

didi`1 “ di`1di ` rdi, di`1s, andd1d2 . . . di´1didi`1di`2 . . . dn “ d1d2 . . . di´1di`1didi`2 . . . dn ` d1d2 . . . di´1rdi, di`1sdi`2 . . . dn.


Thusd1d2 . . . dm ` Vm´1 “ d1 . . . di´1di`1didi`2 . . . dn ` Vm´1.

Since Sympmq is generated by the transpositions pi, i` 1q, 1 ď i ă m this gives

d1d2 . . . dm ` Vm´1 “ dπp1q . . . dπpmq ` Vm´1.

for all π P Sympmq. Choosing π such that bπp1q ď bπp2q ď . . . ď bπpmq and we see d1d2 . . . dm P Vm.

(c) follows from (a) and (b). �

Lemma 1.5.14. Let B be an ordered basis for the Lie algebra L. Identify L with its image in S “ S pLq. Letb P B and s P Bn. Define b ď s if

s “ b1b2 . . . bn with b ď bi P B for all 1 ď i ď n.

Then there exists a unique action ¨ of L on S such that

(i) b ¨ s “ bs for all b P B, n P N and s P Bn with b ď s.

Proof. For m ě 0 put S m :“ xLi | 0 ď i ď myK ď S .To show the uniqueness let ¨ is an action of L on S such that (i) holds. We will show by induction on m

that the restriction of ¨ to Lˆ S m is unique and that

(1˝) wpb, sq :“ b ¨ s´ bs P S m and b ¨ s “ bs` wpb, sq P S m`1 for all b P L, s P S m.

Note here that the first formula implies the second. Also by K-linearity it suffices to show (1˝) for b P Band s P Bm.

So let b P B and s P Bm. We need to compute b ¨ s uniquely and show that b ¨ s´ bs P S m.

Suppose first that b ď s. Then by (i):

(2˝) b ¨ s “ bs pif b ď sq.

So b ¨ s is uniquely determined and wpb, dq “ b ¨ s´ bs “ 0 P S m. So (1˝) holds in this case.

Suppose next that that b ę s. Then m ě 1. Since S is commutative there exist unique d P B and t P Bm´1

with s “ dt and d ď t. Then by (i) d ¨ t “ dt “ s and so

b ¨ s “ b ¨ pd ¨ tq.

Since ¨ is a Lie-algebra action, rb, ds ¨ t “ b ¨ pd ¨ tq ´ d ¨ pb ¨ tq and so

b ¨ pd ¨ tq “ d ¨ pb ¨ tq ` rb, ds ¨ t.

Since wpb, tq “ b ¨ t ´ bt, b ¨ t “ bt ` wpb, tq and so

d ¨ pb ¨ tq “ d ¨ pbtq ` d ¨ wpb, tq.

Since b ę s “ dt and d ď t we have d ă b and d ď bt. Hence (i) gives

d ¨ pbtq “ dbt “ bdt “ bs.


Combing the last four displayed equation gives

(3˝) b ¨ s “ bs` d ¨ wpb, tq ` rb, ds ¨ t pif b ę s, s “ dt, d P B, t P Bm´1, d ď tq

By induction on m, b ¨ t and rb, ds ¨ t are uniquely determined and contained in S m. In particular, wpb, tq “b ¨ t ´ bt is uniquely determined. Also (i) holds for m ´ 1 and so wpb, tq P S m´1. Thus again by inductiond ¨wpb, tq is uniquely determined and contained in S m. So (3˝) uniquely determines b ¨ s. Moreover, wpb, sq “d ¨ wpb, tq ` rb, ds ¨ t P S m. Thus ¨ is unique and (1˝) holds.

To prove existence we define b ¨ s for b P B and s P Bm by induction on m via (2˝) and (3˝). Once b ¨ sis defined for all s P Bm, define l ¨ s for all l P L and s P S m by linear extension. Note also that (1˝) will holdinductively. In particular, wpb, tq P S m´1 and so all terms on the right side of (3˝) are defined at the time theyare used to define the left side.

It remains to verify that ¨ is an action. Let a, b P L and s P S . Just for this proof we say that pa, bq acts ons if

a ¨ pb ¨ sq ´ b ¨ pa ¨ sq “ ra, bs ¨ s.

Note that set of s P S on which pa, bq acts is a K-subspace of V . Also pa, bq acts on s if and only if pb, aqacts on s.

We will now show:

4˝. Let a, b P B and s P Bm with a ď s or b ď s. Then pa, bq acts on s.

If a “ b, this clearly holds. So by symmetry we may assume that a ą b. Then b ď s, a ę bs and b ď as.

a ¨ pb ¨ sq “ a ¨ pbsq ´ b ď s, (2˝)“ abs` b ¨ wpa, sq ` ra, bs ¨ s ´ a ę bs, (3˝)“ abs` b ¨ pa ¨ sq ´ b ¨ pasq ` ra, bs ¨ s ´ definition of wpa, sq

“ abs` b ¨ pa ¨ sq ´ bas` ra, bs ¨ s ´ b ď as, (2˝)“ b ¨ pa ¨ sq ` ra, bs ¨ s

So (4˝) holds.

Suppose inductively that

5˝. For all a, b P L and all s P S m´1, pa, bq acts on s.

We will now that (4˝) also hold m in place of m ´ 1. By K-linearity we may assume that a, b P B ands P Bm, and by (4˝) we may assume that a ę s and b ę s. Thus m ą 0 and s “ dt where d P B and t P Bm´1

with d ď t. Then by (2˝)

(6˝) s “ dt “ d ¨ t.

Since b ę d and d ď t we have d ď b and d ď bt. So (4˝) gives that pa, dq acts on bt. Since wpb, tq P S m´1the induction assumption (5˝) shows that pa, dq acts on wpb, tq P S m´1. Thus

(7˝) pa, dq acts on b ¨ t “ bt ` wpb, tq.


This allows us to compute (using our inductive assumption (5˝) various times):

a ¨ pb ¨ sq

“a ¨ pb ¨ pd ¨ tqq ´ s “ d ¨ t by (6˝)

“a ¨`

d ¨ pb ¨ tq ` rb, ds ¨ t˘

´ pd, bq acts on t by (5˝)

“d ¨ pa ¨ pb ¨ tqq ` ra, ds ¨ pb ¨ tq ` a ¨`

rb, ds ¨ t˘

´ pa, dq acts on b ¨ t by (7˝)“d ¨ pa ¨ pb ¨ tqq ` ra, ds ¨ pb ¨ tq ` rb, ds ¨ pa ¨ tq ` ra, rb, dss ¨ t ´ pa, rb, dsq acts on t by (5˝)

Since the situation is symmetric in a and b the above equation also holds with the roles of a and binterchanged. Subtracting these two equations, and observing that the two middle terms cancel, we obtain

a ¨ pb ¨ sq ´ b ¨ pa ¨ sq

“d ¨`

a ¨ pb ¨ tq ´ b ¨ pa ¨ tq˘

``

ra, rb, dss ´ rb, ra, dss˘

¨ t

“d ¨`

ra, bs ¨ t˘

q ``

ra, rb, dss ` rb, rd, ass˘

¨ t ´ pa, bq acts on t by (5˝)

“ra, bs ¨ pd ¨ tq ` rd, ra, bss ¨ t ``

ra, rb, dss ` rb, rd, ass˘

¨ t ´ pd, ra, bsq acts on t by (5˝)

“ra, bs ¨ s``

rd, ra, bss ` ra, rb, dss ` rb, rd, ass˘

¨ t ´ s “ d ¨ t by (6˝)“ra, bs ¨ s ´ Jacobi identity

Thus pa, bq acts on s and so by induction L acts on S . �

Theorem 1.5.15 (Poincare-Birkhof-Witt). Let pU, f q be an universal enveloping algebra of L. Let B be anordered basis of L and view S pLq as an L- module via 1.5.14. Note that by 1.5.12(a), S pLq is also a modulefor U.

(a)D :“

`

f pb1q f pb2q . . . f pbnq | n P N, b1 ď b2 ď . . . ď bn P B˘

.

is a K basis for U.

(b) The map α : U Ñ S pLq, u Ñ u ¨ 1S is a isomorphism of K-spaces.

(c) f is 1-1.

Proof. Let b1 ď b2 ď . . . bn P B. Then bi´1 ď bi . . . bn and definition of the action of L on S pLq andinduction shows that

f pb1q f pb2q . . . f pbnq ¨ 1 “ f pb1qp f pb2qp. . . p f pbnq ¨ 1qq “ b1b2 . . . bn.

Hence α mapsD to the basis

`

b1b2 . . . bn | n P N, b1 ď b2 ď . . . ď bn P B˘

.

of S pLq. It follows that that α is onto andD is linearly independent in U. By 1.5.13(c), xDyK “ U and soDis a basis for U. So (b) holds. Also α sends a basis of U to a basis of S pLq and thus α is a K- isomorphism.Note that p f pbq | b P Bq is a subfamily of D and so linearly independent. Since B is a basis of L this showthat f is 1-1 and so (c) holds. �


Notation 1.5.16. From now on U “ UpLq denotes a universal envelpong algebra for L. In view of thePoincare-Witt-Birkhoff Theorem we may and do identify L with its image in U. So

L Ď U and`

b1b2 . . . bn | n P N, b1 ď b2 ď . . . ď bn P B˘

is a K-basis for U

We also identify k P K with k1U in U.(We will no longer use the symmetric algebra for L, so this should not lead to confusion with the corre-

sponding conventions for S pLq.)According to 1.5.12 we view every L-module V as an U-module. Note that if a1, a2, . . . , an P L and v P V,

then

pa1a2 . . . anqv “ a1pa2p. . . panvq . . .qq.

In particular, the adjoint action of L on L extends to an action of U on L. We denote this action by

U ˆ L Ñ L, u Ñ u ˚ l.

For example, if a, b, l P L then

a ˚ l “ ra, ls and pabq ˚ l “ a ˚ pb ˚ lq “ ra, rb, lss.

and

L3 ˚ L “ rL, rL, rL, Lsss

1.6 Nilpotent ActionDefinition 1.6.1. (a) Let R be a ring and S Ď R. We say that S is nilpotent if S n Ď 0 for some n P N.

(b) Let L be a Lie algebra and B Ď L. We say that B is nilpotent of Bn ˚ B Ď 0 for some n P N.

(c) Let A be an associative or Lie algebra, V a module for A, B Ď A and W an B-invariant subset of V.We say that B annihilates W if BW Ď 0 and we say that B acts nilpotently on W if BnW Ď 0 for somen P N.

(d) Let A be an associative or Lie algebra, V a module for A, B Ď A and n P N. Define AnnnBpVq

inductively by Ann0VpBq “ 0 and

Annn`1V pBq “ AnnV

`

B,AnnnVpBq

˘

1

Lemma 1.6.2. Let A be an associative or Lie algebra over K, V a module for A and B Ď A.

(a) B acts nilpotently on V if and only if the image of B in EndKpVq is nilpotent.

(b) If A is associative, then B is nilpotent if and only if B acts nilpotently on A by left multiplication.

(c) If A is Lie-algebra, and B acts nilpotently on A by the adjoint action, then B is nilpotent.

(d) If A is Lie-algebra and B is an ideal in A, then B is nilpotent if and only if B acts nilpotently on A bythe adjoint action.

1See 1.4.6(a) for the definition of AnnV pB,Uq

1.6. NILPOTENT ACTION 23

Proof. (a): Let Φ be homomorphism associated to the action of A on V . If a P A, then aV “ 0 if and only ifΦpaq “ 0. Note also that ΦpBnq “ ΦpBqn. So BnV Ď 0 if and only if ΦpBqn Ď 0.

(b): If Bn Ď 0, then also BnA Ď 0. If BnA Ď 0, then Bn`1 Ď 0.(c) If Bn ˚ A “ 0, then also Bn ˚ B “ 0.(d) Suppose Bn ˚ B “ 0. Since B is an ideal, B ˚ A Ď B and so

Bn`1 ˚ A “ Bn ˚ pB ˚ Aq Ď Bn ˚ B “ 0.

�

Example 1.6.3. Consider the Lie algebra L with basis x, y such that rx, ys “ x. Then Ky is an abelian and soa nilpotent subalgebra of L. But y ˚ x “ ´x and so yn ˚ x “ p´1qnx ‰ 0 for n P N . Thus Ky does not actnilpotently on L.

We remark that if X acts nilpotently on V then all elements in X act nilpotently on V . The main goal ofthis section is to show that for finite dimensional Lie-algebras, the converse holds. That is if all elements ofthe finite dimensional Lie-algebra L act nilpotently on V , then also L acts nilpotenly on V .

Lemma 1.6.4. Let A be an associative or Lie algebra. Let V be an A-module and B Ď A. Then the followingare equivalent:

(a) B acts nilpotently on V.

(b) There exists a finite chain of B-invariant K-subspaces 0 “ Vn ď Vn´1 ď . . .V0 “ V such that Bannihilates each each Vi{Vi`1.

(c) V “ AnnnVpBq for some n P N.

(d) There exists a finite chain of B-invariant K-subspaces 0 “ Vn ď Vn´1 ď . . .V0 “ V such that B actsnilpotently on each Vi{Vi`1.

Proof. (a) ùñ (b): Let n P N with BnV Ď 0. and define

Vi “ xBkV | i ď k ď ny

Then V0 “ V , Vn “ 0, Vi is a K-subspace of V and BVi Ď Vi`1 ď Vi. In particular, Vi is B-invariant andB annihilates on Vi{Vi`1.

(b) ùñ (c): Put Wi “ Vn´i. Then BWi`1 Ď Wi.. We will show that Wi ď AnniVpBq. This clearly

holds for i “ 0. Suppose inductively that Wi Ď AnniVpBq. Then BWi`1 Ď Wi Ď Anni

VpBq and so Wi`1 Ď

Anni`1V pBq. Thus V “ V0 “ Wn Ď Annn

VpBq and (d) holds.

(c) ùñ (d): Just observe that B annihilates Anni`1V pBq{Anni

VpBq and so also acts nilpotently onAnni`1

V pBq{AnniVpBq.

(d) ùñ (a): For 0 ď i ă n choose mi with BmipVi{Vi`1q “ 0. Then Bmi Vi ď Vi`1. Put m “řn´1

i“0 mi.Then BmV “ 0. �

Corollary 1.6.5. Let A be an associative or Lie algebra and B Ď A. Let B˚ be the K-subalgebra of Agenerated by B.

(a) Let V be a A-module. Then AnniVpBq “ Anni

VpB˚q for all i P N. In particular, B acts nilpotently on V

if and only if B˚ acts nilpotently on V.


(b) B is nilpotent if and only if B˚ is nilpotent.

Proof. (a) Thus is true for i “ 0. By 1.4.9(g)

AnnV`

B,AnniVpBq

˘

“ AnnV`

B˚,AnniVpBq

˘

and so (a) follows by induction.(b) Suppose that B is nilpotent. If A is associative, B acts nilpotently on A by left multiplication. Hence

by (a) the same is true for B˚ and so B˚ is nilpotent.So suppose A is a Lie-algebra and consider the adjoint action of A on A. Since B is nilpotent, B Ď

AnnnApBq for some n ě 1. Since

AnnnApBq “ AnnA

`

B,Annn´1A pBq

˘

1.4.9(e) show that AnnnApBq is K-subalgebra of Annn

ApBq. Thus B˚ Ď AnnnApBq. By (a) Annn

ApBq “Annn

ApB˚q and so B˚ is nilpotent. �

Lemma 1.6.6. For all n P N : Ln ˚ L Ď xLn`1y.

Proof. The proof is by induction on n. L0 “ t1u and so L0 ˚ L “ L “ xL1y. Let l P L and a P Ln´1 ˚ L. Byinduction, a P xLny and so

l ˚ a “ rl, as “ la´ al P LxLny ` xLnyL Ď xLn`1y.

ThusLn ˚ L “ L ˚ pLn´1 ˚ Lq Ď xLn`1y.

�

Lemma 1.6.7. Suppose L acts nilpotenly on the L-module V. Then L{AnnLpVq is nilpotent.

Proof. Without loss V is faithful L-module. Since L acts nilpotently on V , LnV “ 0 for some n ě 1. By 1.6.6Ln´1 ˚ L Ď xLny and so

pLn´1 ˚ LqV Ď xLnyV “ xLnVy “ 0.

Note that Ln´1 ˚ L Ď L and so Ln´1 ˚ L Ď AnnLpVq “ 0 and L is nilpotent. �

Lemma 1.6.8. Let A be an associative or Lie algebra. Let V be an A-module and D, E Ď A with rE,Ds Ď D.If E and D acts nilpotently on V, then E Y D acts nilpotently on V.

Proof. In the case that A is associative, we replace A by LpAq. So A is now a Lie algebra. Since rE,Ds Ď D,E is left D-invariant. By 1.4.12 xDnVy is E Y D-invariant. Note that D acts annihilates xDnVy{xDn`1Vy andE acts nilpotenly on on xDnVy{xDn`1Vy for all n. Thus D X E acts nilpotently on xDnVy{xDn`1Vy. SinceD acts nilpotently on V , DnV “ 0 for some n P N. Lemma 1.6.4 now show that E Y D acts nilpotently onV . �

Lemma 1.6.9. Let A be an associative K-algebra.

(a) Let D, E Ď A be nilpotent with rE,Ds Ď D. Then DY E is nilpotent.

(b) Let D Ď A be nilpotent. Then D acts nilpotently on LpAq by adjoint action.

1.6. NILPOTENT ACTION 25

Proof. (a) By 1.6.8 DY E acts nilpotently on A and so is nilpotent.(b) Let n P N with Dn Ď 0. Recall that l and r are the representation of A on A by left and right

multiplication, respectively, see 1.3.3. Since Dn Ď 0 we have lpDqn “ 0 and rpDqn “ 0. Since both lpDq andrpDq are nilpotent subsets of EndKpAq. Also since A is associative, lpaq ˝ rpbq “ rpbq ˝ lpaq for all a, b P R.Thus rlpDq, rpDqs Ď 0. Hence (a) implies that lpDq Y rpDq is nilpotent. Thus by 1.6.5(b) also the subalgebraB˚ generated by lpDq Y rpDq is nilpotent. Since ad paq “ lpaq ´ rpaq we have ad pDq Ď B˚ and so ad pDq isnilpotent in EndpAq. Thus D acts nilpotently on LpAq by adjoint action. �

Corollary 1.6.10. Let V be a faithful L-module and suppose that B Ď L acts nilpotently on V. Then B actsnilpotently on L by adjoint action.

Proof. Let Φ : L Ñ glpVq be the corresponding representation. Since B acts nilpotently on V , ΦpBq isnilpotent in EndpVq, see 1.6.2(a). From 1.6.9 the adjoint action of ΦpXq on glpVq is nilpotent. Thus ΦpBqacts nilpotently on ΦpLq. Since Φ : L Ñ ΦpLq is isomorphism of L-modules, B acts nilpotently on L. �

Lemma 1.6.11. Let V be L-module, B Ď L and 0 Ď W Ĺ V. If B acts nilpotenly on V, then AnnVpB,Wq ĘW.

Proof. Since B acts nilpotently on V , BmV Ď 0 Ď W for some m P N. So we can choose n P N minimal withBnV Ď W. Since W ‰ V , n ‰ 0. By minmality of n, Bn´1V Ď W. Then BpBn´1Vq “ BnV Ď W and soBn´1V Ď AnnVpB,Wq. Since Bn´1V Ę W, also AnnVpB,Wq Ę W. �

Corollary 1.6.12. Let B be a subalgebra of L with B ‰ L and suppose B acts nilpotently on L by adjointaction. Then B Ĺ NLpBq.

Proof. Since B is a subalgebra of L, rB, Bs Ď B and so B Ď NLpBq. Note that

NLpBq “ tl P L | rl, Bs Ď Bu “ tl P L | rB, ls Ď Bu “ AnnLpB, Bq

Since 0 Ď B Ĺ L, 1.6.11 shows that AnnLpB, Bq Ę B and so the corollary holds. �

Theorem 1.6.13. Let L be a finite dimensional Lie algebra and V a L-module. If all elements of L actnilpotently on V, then L acts nilpotently on V.

Proof. We may assume without loss that L is faithful on V . The proof is by induction on dim V . Let M be amaximal subalgebra of L. By induction M acts nilpotently on V . So by 1.6.10 M acts nilpotenly on L. 1.6.12implies that there exists d P NLpMqzM. Note that Kd and M ` Kd are subalgebras of L. By maximality ofM, L “ M`Kd ď NLpMq. As d is nilpotent on V , Kd is nilpotent on V as well. Thus 1.6.8 implies M`Kdacts nilpotently on V . As L “ M ` Kd this shows that L acts nilpotently in V . �

Corollary 1.6.14 (Engel). Let L be a finite dimensional Lie algebra all of whose elements act nilpotently onL. Then L is nilpotent.

Proof. Apply 1.6.13 to the adjoint module. �

Definition 1.6.15. Let A be a K-algebra and I Ď A. We write I Ĳ A if I is an ideal in A. We say that I is asubideal in A and write I ĲĲ A if there exists chain I “ I0 Ĳ I1 Ĳ . . . Ĳ In Ĳ In “ A.

Lemma 1.6.16. Suppose L is nilpotent. Then every subalgebra in L is an subideal in L.

Proof. Let n be minimal with Ln ˚ L “ 0 and A ď L. Let Z “ Ln´1 ˚ L. Then L ˚ Z “ 0, that is rL,Zs “.Thus rA,Z`As “ rA, As Ď A and A Ĳ Z`A. Put L “ L{Z. Since Ln´1 ˚L ď Z, L

n´1˚L “ 0. By induction

on n we may assume Z ` A{Z ĲĲ L{Z. Thus Z ` A ĲĲ L and so A ĲĲ L since A Ĳ Z ` A ĲĲ L �


1.7 Finite Dimensional ModulesDefinition 1.7.1. Let A be Lie or an associate K-algebra and let V and W be A-modules.

(a) A function f : V Ñ W is called A-linear if f is K-linear and f pavq “ f pavq for all a P A and v P V. fcalled a A-isomorphism if f is A-linear bijection.

(b) We say that V and W are isomorphic A-modules if there exist a A-isomorphism f : V Ñ W.

(c) V is called A-simple if V has no proper A-submodules, that is 0 and V are the only A-submodules.

(d) V is called A-semisimple if V is the direct sum of simple A-submodules.

(e) V is called A- homogeneous if its the direct sum of isomorphic simple A-submodules.

(f) If X and Y are A-submodules of V with X ď Y, then Y{X is called an A-section of V.

Definition 1.7.2. Let A be an associative or Lie-algebra and V an A-module.

(a) An A- series on V is a chain S of A-submodules of V such that

(i) 0 P S and V P S.

(ii) S is closed under intersections, that isŤ

D P S for allH ‰ D Ď S.

(iii) S is closed under unions, that isŞ

D P S for allH ‰ D Ď S.

(Here a chain is a set of sets which is totally ordered with respect inclusion)

(b) Let S be an A-series. A jump of S is pair pB,T q such that

(i) B,T P S.

(ii) B ň T.

(iii) If D P S with B ď D ď T then D “ B or D “ T.

If pB,T q is a jump of S, then T{B is called a factor of S.

(c) An A-composition series on V is a A-series on V all of whose factors are simple A-modules.

(d) Let S and T be A-series on V. We say that S and T have isomorphic factors if there exists a bijectionΦ between the sets of factors of S and T such that for each factor F of S, F and ΦF are isomorphicA-modules. Such a Φ is called an isomorphism of the sets of factors.

Remark 1.7.3. A finite A-series on V is just a sequence

0 “ V0 ă V1 ă V2 . . . ă Vn´1 ă Vn “ V

of A-submodules of V. The jumps of this series are the pairs pVi´1,Viqq, 1 ď i ď n and the factors areVi{Vi´1. Note that if V is finite dimensional over K, every A-series if finite.

Lemma 1.7.4. Let A be an associative or Lie-algebra, V be an A-module and W an A-submodule of V. LetS be a A-series on W and T a A-series on V{W. Put

T “ tT | W ď T ď V,T{W P T u

Then

1.7. FINITE DIMENSIONAL MODULES 27

(a) SY T is a series for A on V.

(b) The factors of SY T are the factors of S and the factors of T .

(c) S Y T is an A-composition series on V if and only if S is an A-composition series on W and T areA-composition series on V{W.

Proof. This follows readily from the definition. We leave the details as an exercise. �

Lemma 1.7.5 (Jordan Holder). Let A be a Lie or an associative K-algebra. Suppose that there exists a finitecomposition series for A on V. Then any two composition series for A on V are finite and have isomorphicfactors.

Proof. Let S be a finite composition series for A on V and T any compostion series. For a jump pB,Cq of Tchoose D P S maximal with C ę B ` D. Let E be minimal in S with D ă E. Then E{D is a factor of T ,C{B is a factor of S and we will show that map B{C Ñ E{D is an isomorphism of the sets of factor.

By maximality of D we haveC ď B` E.

Thus C “ C X pB` Eq “ B` pC X Eq and so

C{B – C X E{C X E X B “ C X E{BX E.

Since C ę B` D, C X E ę D and since E{D is simple, E “ D` pC X Eq. Thus

E{D – C X E{C X D.

If BX E ę D, then E “ pBX Eq ` D ď B` D and so C ď B` E ď B` D, contrary to our choice of D.Thus BX E ď D and so BX E “ BX D. Suppose that C X D ę B. Then C “ pC X Dq ` B ď B` D, againa contradiction. Thus C X D “ BX D “ BX E and so

C{B – C X E{BX D “ C X E{BX E – E{D.

It remains to show that our map between the factor sets is a bijection. Let pB1,C1q be a jump other thanpB,Cq and say C1 ď B. Then C1 X E ď BX E “ BX D ď D and so pB1,C1q is not mapped to E{D. So ourmap is one to one.

Since S is finite we conclude that, T has finitely many jumps and so also T is finite and |T | ď |S|. Butnow the situation is symmetric in T and S. Thus |S| ď |T |, |S| “ |T | and our map is a bijection. �

Definition 1.7.6. Let V be an abelian group, and D a set of non-trivial subgroups of V. We say that Dindependent if

ř

D “À

D, that is

D Xÿ

D‰EPD

E “ 0

for all D P D.

Lemma 1.7.7. Let A be an associative or Lie-algebra, V an A-modules andD a set of simple A-submodulesin V. Suppose that V “

ř

D.

(a) Let W be an A-submodule of V, Then there exists E Ď D such that V “ W ‘À

E.

(b) Let W be an A-submodule of V. Then V “ W ‘ Z for some A-submodule Z if V.


(c) Let X be an A-section of V. Then there exists E Ď D with X –À

E as an A-module.

(d) Every A-section of V is a semisimple A-module. In particular, V is a semisimple A-module.

(e) Every composition factor of A on V is isomorphic to some member ofD.

Proof. (a): Let I be the set of independent subsets E of D with W Xř

E “ 0. Order I by inclusion. LetJ be chain in I and put E “

Ť

J . We claim that E P I. Let D P E and

d P DXÿ

D‰EPE

E

Then there exists a finite subset F of EztDu with d Př

F . Since J is a chain and E “Ť

J there existsG P I with F Ď G. Hence

d P DXÿ

D‰EPG

E

and since G P I, the latter intersection is 0. So d “ 0 and D Xř

D‰EPEEE is independent. A similarargument shows that W X

ř

E “ 0 and so E P I.So every chain in I has an upper bound. By Zorn’s Lemma, I has a maximal elements E. Suppose

that V ‰ W `ř

E. Then there exists D P D with D ę W `ř

E. Since D is a simple A-module,DXpW`

ř

Eq “ 0. But then EYtDu is independent and WXpD`ř

Eq “ 0. Hence EYtDu is containedin I. Thus V “ W `

ř

E. Since E is independent and W Xř

E “ 0 we get

V “ W `ÿ

E “ W ‘ÿ

E “ W ‘à

E

and so (a) holds.

For later use note that V{W –À

E and so

1˝. Let W be a A-subspace of V, then V{W –À

E for some E Ď D.

(b) Apply (a) and put Z “À

E.

(c) Let X “ W{R for some A-submodules W and R of V . By (c) V “ W ‘ Z for some A-submodule Zof V . Hence W – V{Z and by (1˝), V{Z –

À

F for some subset F of D. Thus also W –À

F . Now (1˝)applied to pR,W,F q in place of pW,V,Dq shows that there exists a subset E of F with X “ W{R –

À

E asan A-module.

(d) and (e) follow directly from (c). �

Definition 1.7.8. Let A be a associative or Lie-Algebra.

(a) NilpAq is the sum of all the nilpotent ideals of A.

(b) Let V an A-module. Then NilApVq :“Ş

tAnnApWq | W a simple A-section of Vu

Lemma 1.7.9. Let A be a associative or Lie-Algebra, V an A-module and B a right ideal in A. Suppose Bacts nilpotently on V.

(a) If V is a simple A module, then B ď AnnApVq.

(b) B ď NilApVq.


Proof. (a): Since B acts nilpotently on V and V ‰ 0, also AnnVpBq ‰ 0. Since B is a right ideal in A,1.4.9(b) shows that AnnVpBq is an A-submodule of V . As V is a simple A-module this gives V “ AnnVpBqand so B ď AnnApVq.

(b): Let W be a simple A-section of V . Since B acts nilpotently on V , B acts nipotently on W and so (a)shows that B ď AnnApWq. By definition

NilApVq “č

tAnnApWq | W a simple A-section of Vu

and so B ď AnnApVq �

Lemma 1.7.10. Let A be a associative or Lie-Algebra and V an A-module. Suppose S a finite compositionseries for A on V.

(a) NilApVq “ tAnnApFq | F a factor of Su.

(b) NilApVq acts nilpotently on V. In particular, NilApVq is the largest right ideal of A acting nilpotentlyon V.

Proof. (a): Let W “ X{Y be simple A-section of V and let T be a composition series for A on V withX,Y P T . Then W is a factor of V and so by the Jordan Holder Theorem 1.7.5 W is isomorphic to a factor Fof S. Then AnnApFq “ AnnApWq and so It follows that

č

tAnnApFq | F a factor of Su ďč

tAnnApWq | W a simple A-section of V “ NilApVq

Since all factors of S are simple, the reverse inclusion is obvious. So (a) holds.

(b): Since S is finite, S “ tVi | 0 ď i ď nu with

0 “ V0 ă V1 ă V2 ă . . . ă Vn´1 ă Vn “ V,

and factors Vi{Vi´1, 1 ď i ď n. It follows that NilApVq annihilates each Vi{Vi´1 and so by 1.6.4 NilApVq actsnilpotently on V . By 1.7.9 each right ideal of A acting nilpotently on V is contained in NilApVq and so (b)holds. �

Proof. NilLpVq is the intersection of ideals and so an ideal in L. Let

0 “ V0 ă V1 ă . . . ă Vn´1 ă Vn “ V

be a composition series for L on V . Then NilLpVq acts trivially on all the factor and so by 1.6.4(b), NilLpVqacts nilpotently on V .

Conversely, let I be an ideal of L acting nilpotently on V . Let W be a composition factor for L on V . SinceI acts nilpotently in V , I acts nilpotently on W. Hence AnnWpIq ‰ 0. Since I is a ideal, AnnWpIq is a L-submodule of W, see 1.4.9(c) Since W is a simple L-module this gives AnnWpIq “ W. Hence I ď AnnLpWqand I ď NilLpVq. �

Corollary 1.7.11. View L as an L-module via the adjoint action.

(a) NilpLq ď NilLpLq.

(b) Suppose there exists a finite L-compositions series for L on L. Then NilpLq “ NilLpLq and NilpLq isnilpotent. In particular, NilpLq is the largest nilpotent ideal of L.


Proof. By 1.6.2(d) An ideal in L is nilpotent if and only if it acts nilpotently on L. By 1.7.9(b) each idealacting nilpotently on L is contained in NilLpLq. So NilpLq ď NilLpLq and (a) holds. Moreover, if there existsa finite L-compositions series for L on L, then 1.7.10 shows that NilLpVq acts nilpotently on V . So NilpVq isnilpotent and NilLpVq ď NilpVq. �

The following example shows there may not exist a unique largest nilpotenly acting subideal in L:

Example 1.7.12. Let L “ slpK2q, V “ K2 and

x “ E12 “

»

–

0 1

0 0

fi

fl , y “ E21 “

»

–

0 0

1 0

fi

fl and h “ E11 ´ E22 “

»

–

1 0

0 ´1

fi

fl

Note that x2 “ 0 “ y2 and so both Kx and Ky act nilpotently on V .Also

rh, xs “ 2x, ry, hs “ 2y and rx, ys “ h.

In particular, L “ xx, y, hyK “ xx, yyLie is the K-subalgebra of L generated by x and y.If charK “ 2, we have

rh, xs “ 0, ry, hs “ 0 and rx, ys “ h.

Thus

Kx Ĳ Kx` Kh Ĳ L and Ky Ĳ Ky` Kh Ĳ L

and Kx and Ky are subideals acting nilpotently on V . Since LV “ V , L does not act nilpotently on V . ThusKx and Ky are not contained in common nilpotently acting subideal of L.

Notation 1.7.13. Let A be an associative or Lie-algebra. Then SimpLq is the class of all simple A-modules.D Ď SimpLq means thatD is a class of simple L-modules closed under isomorphism. So each member ofDis a simple A-module and if D P D and E is A-module isomorphic to D then E P D.

Definition 1.7.14. Let A be an associative or Lie-algebra, V an A-module andD Ď SimpAq.

(a) V is called aD-module for A if all simple A-sections of V are contained inD.

(b) VD is the sum of all the simpleD-submodules of V for L.

(c) For n P N define VDpnq inductively by VDp0q “ 0 and

VDpn` 1q{VDpnq :“`

V{VDpnq˘

D.

(d) VDp8q :“Ť8

i“0 VDpiq.

(e) VDp˚q is A-submodule of V generated by all theD-submodules of V for A.

(f) Let A ď L andA Ď SimpAq. ThenA|L is the class of simple L-modules which areA-modules for A.


Example 1.7.15. Let L be the subalgebra of glpK3q consisting of all 3ˆ 3 matrices of the form»

—

—

—

–

0 ˚ ˚

0 ˚ ˚

0 0 0

fi

ffi

ffi

ffi

fl

.

Let V “ K3 viewed as an L-module via left multiplication. Let pe1, e2, e3q be the standard basis for K3.For 0 ď i ď 3 put Vi “

řij“1 Ke j. Observe that Vi is the unique i-dimensional L-submodule of V and

0 “ V0 ă V1 ă V2 ă V3 “ V.

is the unique composition series for L on V . Put Ik “ Vk{Vk´1. Then Ik is a simple 1-dimensional L-module.Note that LI1 “ 0 and LI3 “ 0 while LI2 ‰ 0. So I1 – I3 but I1 fl I2 as L-module. For k “ 1, 2 letDk be theisomorphism class of Ik. Also letD “ D1 YD2.

By definition, VD1 is the sum of all the simple L-submodule of V contained in isomorphic to D1, that isthe sum of all the simple L-submodules of V isomorphic to I1. As V1 is the only simple L-submodule of Vand V1 – I1 we get VD1 “ V1. To compute VD1 , put V “ V{V1. The only simple L- submodule of V isI2 “ V2{V1. Since I1 fl I2 we get VD1 “ 0. Thus VD1p2q “ V1. It follows that VD1p jq “ V1 for all j ě 1 andso also VD1p8q “ V1.

No submodule of V is isomorphic to I2 and hence VD2 “ 0. Thus VD2p8q “ VD2p jq “ 0 for all j ě 0.

V1 is the only submodule of V isomorphic to I1 or I2 and so VD “ V1. V2{V1 is the only submodule ofV{V1 isomorphic to I1 or I2. So pV{V1qD “ V2{V1 and VDp2q “ V2. V{V2 “ I3 is isomorphic to I1 and soV “ VDp3q “ VDp8q.

Lemma 1.7.16. Let A be an associative or K-algebra,D Ď SimpAq and V an A-module.

(a) If V is a simple A-module, then V is aD-module for A if and only if V P D.

(b) Let W be an A-submodule of V. Then V is anD-module for A if and only if W and V{W areD-modulesfor A.

(c) Let S be a finite A-series on V. Then V isD-module for L if and only if each factor of S is aD-modulefor A.

(d) VD is the unique maximal semisimpleD-submodule for A in V.

(e) VDp˚q is an D-module for A. In particular, an A-submodule W of V is an D-module for A if and onlyif W ď VDp˚q.

(f) VDp8q Ď VDp˚q. In particular, VDp8q isD-module for A.

(g) Let E Ď SimpLq. Then

VDp˚q X VEp˚q “ VDXEp˚q and VDp˚q ` VEp˚q ď VDYEp˚q.

Proof. (a) : Just observe that V{0 is the only non-zero section of V .

(b): Any A-section of W and any A-section of V{W is isomorphic to an A-section of V . So if V is aD-module for A, so are W and V{W.


So suppose W and V{W are D-modules for A. Let Y{X be a simple A-section of V . Suppose first thatW X Y ď X. Then Y X pX `Wq “ X ` pY XWq “ X and so

Y `W{X `W – Y{`

Y X pX `Wq˘

“ Y{X.

Thus Y{X is isomorphic to a section of V{W and so is contained inD.Suppose next that W X Y ę X. Then X ň pW X Yq ` X ď Y and since Y{X is a simple A-module,

Y “ pW X Yq ` X. Hence

Y{X “ pW X Yq ` X{X – pW X Yq{pW X Y X Xq

and so Y{X is isomorphic to a section of W and so is contained inD.

(c): Follows from (b) and induction on |S|.

(d) Let V be the set of simple D-submodules of V for A. By definition VD “ř

V and by 1.7.7(d) anysum of simple submodules is semisimple. So VD is semisimple. By 1.7.7(e) any L-composition factor ofř

V is isomorphic to some member ofV and so is aD-module for L. Hence VD is aD-module.Conversely, let W be a semisimple D-module submodule for A of V . Then W is a sum of simple A-

submodules. By (b) any L submodule ofD-module is aD-module for L. So each of the simple summand ofW is a simpleD-module and thus W is contained in VD.

(e) Let V be set of D-submodules of V for A. Let W,Z P V. Note that W ` Z{Z – W{W X Z and by(b) W{W X Z is aD-module. Since also Z is aD-module, another application of (b) shows that Z `W isDsubmodule. Hence Z `W P V. It follows that

Ť

V is an additive subgroup of V and

VDp˚q “ÿ

V “ď

V.

Let Y{X be a simple A-section of VDp˚q and let y P YzX.. Since VDp˚q “Ť

V there exists W PW withy P W. Then W X Y ę X, Y “ pW X Yq ` X and so Y{X – W X Y{W X X X Y is isomorphic to a simplesection of W. Since W is aD-module this gives Y{X P D and VDp˚q is aD-module. So the first statement in(e) holds. Since any submodule of aD-module is aD-module, the first statement implies the second.

(f): By (d) VDpn` 1q{VDpnq “ pV{VDpnqqD is aD-module for V . So by (c), VDpnq is aD-module foreach n P N. Thus VDp8q ď VDp˚q. By (e) this implies that VDp8q isD-module for A.

(g) Let W be an A-submodule of V . By (e), W ď VDXEp˚q if and only if W is a D X E-module for A.Note that this is the case if and only if W is D-module and a E-module for A, and so, applying (e) two moretimes, if and only if W ď VDp˚q X VEp˚q. Hence

VDXEp˚q “ VDp˚q X VEp˚q.

Any D-module and any E module for A is an D Y E-module for A. By definition, VDp˚q is a sum ofD-modules and VEp˚q is a sum of E-modules. Hence VDp˚q ` VEp˚q is a sum of D Y E-modules and socontained in VD Y Ep˚q.

�

Lemma 1.7.17. Let rKs be the class of 1-dimensional L-modules annihilated by L and V a finite dimensionalL-module. Then V is a rKs-module if and only if L acts nilpotently on V. In particular, VrKsp˚q is the largestL-submodule of V on which L acts nilpotently.


Proof. Suppose first that L acts nilpotently on V and let W be simple L-section of V . Then L acts nilpotentlyin W and so AnnWpLq ‰ 0. Since W is simple this gives W “ AnnWpLq and W is 1-dimensional. HenceW P rKs and so V is a rKs-module.

Suppose next that V is a rKs-module and let S be a composition series for L on V . Then all factors of Sare in rKs and so annihilated by L. Since V is finite dimensional S is finite and so 1.6.4 shows that L actsnilpotently in V .

By 1.7.16 VrKsp˚q is the largest rKs-submodule of V for L and so the largest L-submodule of V on whichL acts nilpotently. �

Lemma 1.7.18. A be an associative or Lie-algebra Let V be an A-module and D Ď SimpAq. Suppose alsothat there exists S is a finite composition series S for A on V.

(a) V isD-module for A if and only if each factor of S is contained inD.

(b) Let B be a subalgebra of A and B Ď SimpBq. Then V is a B-module for B if and only if V is anB|A-module for A.

(c) VDp8q “ VDp˚q

(d) Let B and E be subalgebras of A with E Ď B. Let E Ď SimpEq and suppose that V is finite dimensionalover K. Then V is an E|B|A-module for A if and only if V is an E|A-module for A.

Proof. (a): By 1.7.16(c), V is D-module for A if and only if each factors of S is D-module for A. SinceS is a composition series, the factors of S are simple. So by 1.7.16(a) the factors of S areD-module for A ifand only if it is contained inD.

(b): By (a) V is B|A-module if and only if each factor of S is B|A-module. By definition of BA, this isthe case of and only if each factor of A is a B-module for B. As S is a finite B-series on V , 1.7.16(c) showsthat this is the case if and only if V is a B-module for B.

(c): By 1.7.16(e), VDp8q Ď VDp˚q. By the Jordan Holder Theorem any composition series for A on Vis finite. It follows that there exists a finite composition series

0 “ W0 ă W1 ă . . . ă Wn “ VDp˚q

for A on VDp˚q. Since VDp˚q isD-module for A, each Wi`1{Wi is contained inD.We claim that Wi ď VDpiq. For i “ 0 this is obvious. So suppose Wi ď VDpiq. Since Wi is a maximal

submodule of Wi`1,Wi`1 X VDpiq “ Wi or Wi`1 X VDpiq “ Wi`1.

In the latter case, Wi`1 ď VDpiq ď VDpi` 1q. In the former, put V “ V{VDpiq. Then

Wi`1 “`

Wi`1 ` VDpiq˘

{VDpiq – Wi`1{Wi`1 X VDpiq “ Wi`1{Wi.

Thus Wi`1 is a simple D-module for L and Wi`1 ď`

V˘

D. Hence the definition of VDpi ` 1q implies

Wi`1 ď VDpi` 1q.So the claim holds. In particular, VDp˚q “ Wn ď VDpnq ď VDp8q and (c) is proved.

(d): Since V is finite dimensional, any A and any B series on V is finite. So we can apply (b) withpA, B,E|Bq, pB, E,Eq and pA, E,Eq in place of pA, B,Bq. Thus


V is a E|B|A-module for A

ðñ V is an E|B-module for B.

ðñ V is an E-module for E.

ðñ V is an E|A-module for A.

�

Lemma 1.7.19. Let L be a Lie algebra, V an L-module and a P L. Let B a K-subalgebra of L, and W aB-submodule of V.

(a) Let X be an I submodule of V containing rB, asW. Then the function

f : W Ñ V{X, w Ñ aw` X

is B-linear.

(b) Suppose B is an ideal of L. Then the function

g : W Ñ V{W, w Ñ aw`W

is B-linear.

(c) Suppose rB, as “ 0. Then the function

h : W Ñ V, w Ñ aw

is B-linear.

Proof. (a) Let b P B and w P W. Then rb, asw P rB, asW Ď X and so

bpawq “ apbwq ` rb, asw P apbwq ` X.

Thusf pbwq “ apbwq ` X “ bpawq ` X “ bpaw` Xq “ b f pwq.

(b) Since B is an ideal, rB, as Ď W and so since W is an B-submodule, rB, asW Ď W. Thus we can apply(a) with X “ W.

(c) Just apply paq with X “ 0.�

Lemma 1.7.20. Let V an L-module, B be an ideal of L and B Ď SimpBq.

(a) Suppose B ď ZpLq and let i P N. Then VBpiq is an L-submodule of V.

(b) VBp˚q is a B|L-submodule of V for L. In particular, VBp˚q ď VB|Lp˚q.

(c) If there exists a finite L-composition series on V, then VBp˚q “ VB|Lp˚q.


Proof. (a): Let a P L and let W be a simple B-submodule of V with B P B. Since B ď ZpLq, rB, ls “ 0 andso by 1.7.19(c) the function h : W Ñ V,w Ñ aw is B-linear. Since W is a simple B-module, either ker h “ 0or ker h “ W. Hence aW “ f pWq – W or aW “ f pWq “ 0. In both cases aW ď VB. Since this holds for allsuch a and W, LVB ď VB and VB is an L-submodule of V . The definition of VBpnq and induction on n nowshows that (a) holds.

(b) Put W “ VBp˚q. We will first show that W is a L-submodule of V .Let a P L. Then by 1.7.19(b), g : W Ñ V{W,w Ñ aw `W is B-linear. Hence gpWq – W{ ker g as an

B-module. By 1.7.16(d), W is a B-module for B and by 1.7.16(b) also W{ ker g is a B-module for B. SincegpWq “ aW ` W{W we conclude that aW ` W{W is B-module for B. The same is true for W and so by1.7.16(b) aW `W is an B-module for B. Since W “ VBp˚q this gives aW `W ď W. Thus aW ď W and Wis an L-submodule of V .

Since W is aB-module for W, any B-section of W is aB-module for B. In particular, any simple L-sectionof W is a B-module for L and so contained in B|L. Thus W is a B|L-module.

(c) By (b) VBp˚q ď VB|Lp˚q. By 1.7.16(d) VB|Lp˚q is an B|L-module for L. Hence 1.7.18(b) shows thatVB|Lp˚q is an B-module for B. Thus VB|Lp˚q ď VBp˚q. �

Lemma 1.7.21. Let V be a finite dimensional L-module, B ĲĲ L and B Ď SimpBq. Then VBp˚q “ VB|Lp˚q.In particular, VBp˚q is an L-submodule for V.

Proof. Note first that since V is finite dimensional, any composition series on V for any subalgebra of L isfinite.

Since B ĲĲ L we can choose a subideal series

B “ B0 Ĳ B1 Ĳ . . . Ĳ Bn´1 Ĳ Bn “ L,

and put E “ Bn´1. By induction on n,VBp˚q “ VB|E p˚q.

Since E Ĳ L, 1.7.20(c) shows that

VB|E p˚q “ VB|E |Lp˚q.

By 1.7.18(d), an L-submodule of V is a B|E |L-module for L if and only if it is a B|L-module for L. Bydefinition, VDp˚q is the sum ofD-submodules of V and so

VB|E |Lp˚q “ VB|Lp˚q.

Combining the last three displayed equation gives proves the lemma. �

Proposition 1.7.22. Let B ĲĲ L and V a finite dimensional simple L-module. Then any two simple B-sectionof V are isomorphic. If in addition B ď ZpLq, then V is an homogeneous B-module.

Proof. Since V is finite dimensional, there exists a simple B-submodule W of V . Let B be the isomorphismclass of W. Then by 1.7.20(c) VBp˚q is a non-zero L-submodule of V . Since V is simple, V “ VBp˚q.Similarly, if B ď ZpLq, then by 1.7.20(b) VB is a non-zero L-submodule and so V “ VB �

Lemma 1.7.23 (Schur). Let V be a simple L-module. Let K: be the image of K in EndKpVq and put D :“EndLpVq. Then

(a) D is a division ring with K: ď ZpDq.

(b) Suppose K is algebraically closed and V is finite dimensional over K. Then D “ K:.


Proof. (a) It easy to see that K: ď ZpDq. Let 0 ‰ d P D. Then V ‰ AnnVpdq is L-submodule of V and sinceV is simple, AnnVpdq “ 0. So d is 1-1. Similarly V “ dV and so d is onto. Simple calculations show thatd´1 P D. So D is a division ring.

(b) Suppose now that V is finite dimensional and K is algebraically closed. Then also EndKpVq and D arefinite dimensional. Since K ď ZpDq and D is division ring, Krds is a finite field extension of K and so d P K.Thus K “ D. �

Lemma 1.7.24. Let L be an abelian Lie algebra and V a simple L-module. Put D “ EndLpVq. Then

(a) D is a field.

(b) V is 1-dimensional over D .

(c) D is generated by K: and L: as a ring, where K: and L: are the images of K and L in EndKpVq.

(d) If K is algebraically closed and V is finite dimensional, then K: “ D and V is 1-dimensional over K.

Proof. Note that L: is abelian and so L: ď D and L: ď ZpDq. Let E be the subring of D generated by K:

and L. Note that E ď ZpDq. Let 0 ‰ v P V . Then Ev is an L-submodule of V and since V is simple, we getV “ Ev. Moreover, if d P D, then dv “ ev for some e P E. Then pd ´ eqv “ 0. Since D is division ring thisgives d “ e and so E “ D. As E ď ZpDq, this shows that D is a field. Also V “ Ev “ Dv and thus V is1-dimensional over D.

Suppose in addition that K is algebraically closed and V is finite dimensional. Then D is a finite extensionof K: and so D “ K:. �

Lemma 1.7.25. Let V be an L-module and ∆ a partition of SimpLq.2 Then`

VDp˚q | D P ∆˘

is independentin V.

Proof. LetD P ∆ and put

W :“ÿ

tVBp˚q | D ‰ B P ∆u.

We need to show that VDp˚q XW “ 0. For this put E “Ť

D‰BP∆ B. Then W ď VEp˚q andDX E “ H.Hence

VDp˚q XW ď VDp˚q X VEp˚q1.7.16(g)“ VDXEp˚q “ VHp˚q.

�

2So each member of ∆ is a class of simple L-module closed under isomorphism and each simple L-module is contained in a uniquemember of ∆

Chapter 2

The Structure Of Standard Lie Algebras

2.1 Solvable Lie Algebras

Definition 2.1.1. (a) Define Lp0q :“ L and inductively,

Lpn`1q :“ xrLpnq, Lpnqsy.

(b) L is called solvable if Lpkq “ 0 for some k P N. The smallest such k is called the derived length of L.

(c) The series

L ě L1 ě Lp2q ě . . . ě Lpkq ě Lpk`1q ě . . . ě8č

n“0

Lpnq ě 0

is called the derived series of L.

(d) SolpLq is the sum of all the solvable ideals of L.

Lemma 2.1.2. Let A and B be ideals in L.

(a) xrA, Bsy is an ideal in L.

(b) Apnq is an ideal in L.

Proof. (a) View L is an L-module via the adjoint action. Note that A is L-invariant and that B is an L-invariantK-subspace of L. So by 1.4.12 also xrA, Bsy is an L-invariant K-subspace of L. So (a) holds.

(b) follows from (a) and induction on n. �

Lemma 2.1.3. (a) Let B Ĳ L. Then L is solvable if and only if B and L{B are solvable.

(b) Let A, B ď L with A ď NLpBq. Then A` B is solvable if and only if A and B are solvable.

(c) Suppose that L is finite dimensional. Then SolpLq is solvable. In particular, SolpLq is the uniquemaximal solvable ideal of L.

37

38 CHAPTER 2. THE STRUCTURE OF STANDARD LIE ALGEBRAS

Proof. (a) If Lpkq “ 0, then Bpkq “ 0 and pL{Bqpkq “ 0. If Bpnq “ 0 and pL{Bqpmq “ 0, then Lpmq Ď B andLpm`nq “ Lpmqpnq Ď Bpnq “ 0.

(b) Suppose A and B are solvable. Since A ď NLpBq, B Ĳ A ` B. By (a) A{A X B is solvable. Hence Band A` B{B – A{AX B are solvable and so by (a) A` B is solvable.

(c) Since L is finite dimensional, there we can choose an solvable ideal B of L with dimK B maximal amaximal solvable. Let A be any solvable ideal in L. Then by (b), A` B solvable ideal and so by maximalityof dimK B, A ď B. Hence SolpAq ď B ď SolpAq and B “ SolpAq. �

Lemma 2.1.4. (a) Lpkq Ď xLk ˚ Ly for all k P N.

(b) Any nilpotent Lie algebra is solvable.

(c) If NilpLq “ 0, then SolpLq “ 0.

Proof. (a) By induction on k: For k “ 0, both sides are equal to L. Suppose Lpkq Ď Lk ˚ L. Then

Lpk`1q “ xrLpkq, Lpkqsy Ď xrL, Lpkqsy Ď xrL, xLk ˚ Lysy “ xrL, Lk ˚ Lsy “ xLk`1 ˚ Ly.

(b) follows from (a).

(c) Suppose SolpLq ‰ 0 and let k be maximal with A :“ SolpLqpkq ‰ 0. Then rA, As Ď SolpLqpk`1q “ 0and so A is nilpotent. By 2.1.2(b), A is an ideal of L and so A ď NilpLq and NilpLq ‰ 0. �

Definition 2.1.5. (a) L1 :“ Lp1q. L1 is called the derived subalgebra of L.

(b) L is called perfect of L “ L1.

(c) Lp˚q is the sum of the perfect ideals in L.

Remark 2.1.6. (a) Lp˚q is perfect and so the unique maximal perfect ideal in L.

(b) If L is finite dimensional, then there exists k P N with Lpkq “ Lpk`1q. It follows that Lp˚q “ Lpkq,L{Lp˚q is solvable, Lp˚q is the unique ideal minimal such that L{Lp˚q is solvable and Lp˚q is the uniquemaximal perfect subalgebra in L.

Definition 2.1.7. Let V be a finite dimensional L-module and let U be the univeral enveloping algebra of L.For u P U define trVpuq “ trpu:q, where u: is the image of u in EndKpVq and trpu:q is the trace of u:. trV

denotes the corresponding function U Ñ K, u Ñ trVpuq. For B Ď U, trBV denotes the restriction of trV to B.

Lemma 2.1.8. Let V be a finite dimensional L-module.

(a) trV is K-linear and trVpabq “ trVpbaq for all a, b P U.

(b) trVplq “ 0 for all L1

(c) LetW be the set of factors of some L-series on V. Then

trV “ÿ

WPW

trW .

(d) If W is an L-module isomorphic to V, then trV “ trW .

Proof. This follows from elementray facts about traces of linear maps. �

2.1. SOLVABLE LIE ALGEBRAS 39

Definition 2.1.9. We say that K is standard if char K “ 0 and K is algebraicly closed. We say that a K-spaceV is standard if K is standard and V is finite dimensional. L is called standard if L is standard as a K-space.

Proposition 2.1.10. Let V be a simple, standard L-module.

(a) rSolpLq, Ls Ď SolpLq X L1 ď SolpLq X ker trV “ SolpLq X AnnLpVq.

(b) The elements of SolpLq act as scalars on V, that is, for each a P SolpLq there exist k P K with av “ kvfor all v P V.

Proof. By 2.1.8(b), L1 ď ker trV and so

(1˝) rSolpLq, Ls ď SolpLq X L1 ď SolpLq X ker trV .

It remains to show that SolpLq X ker trV “ SolpLq X AnnLpVq and the elements of SolpLq act as scalarsof V . Observe that

`

SolpLq ` AnnLpVq˘

{AnnLpVq ď Sol`

L{AnnLpVq˘

.

Hence replacing L by L{AnnLpVq we may assume that V is a faithful L module. Next we show

2˝. Let A be an abelian ideal in L. Then AX ker trV “ 0, dimK A ď 1, A ď ZpLq and the elements of A actsas scalars on V.

Let Z be simple A-submodule of V . Since A is abelian and K is algebraically closed, 1.7.24 implies that Zis 1-dimensional overK. Thus there exists aK-linear function λ : A Ñ Kwith az “ λpaqz for all a P A. SinceA Ĳ L and V is a simple L-module, 1.7.22 show that two simple A-section for A on V . Put m :“ dimK V andletW be the set of factors of some composition series of L on V . Then each member ofW is isomorphic toZ. Since Z is 1-dimensional we get |W| “ m. Thus by 2.1.8

trVpaq “ÿ

WPW

trWpaq “ m ¨ trZpaq “ mλpaq.

As charK “ 01 mλpaq “ 0 if and only if λpaq “ 0. Hence

ker λ “ AX ker trV .

Since both A and LXker trV are ideals in L we conclude that ker λ is an ideal in L. Hence AnnVpker λq is an L-submodule of V . Since V is a simple L-module and 0 ‰ Z ď AnnVpker λqwe conclude that AnnVpker λq “ V .As V is faithful this gives ker λ “ 0. In particular, dim A “ dim Im λ ď 1 and A X ker trV “ 0. AsrA, Ls ď A X ker trV this gives A ď ZpLq and A X L1 “ 0. Hence 1.7.22 shows that V is a homogeneous Amodule and so a direct some of copies of Z. Thus a P A acts as λpaqidV on V .

(a): If SolpLq X ker trV ‰ 0, then the last non-trivial term of the derived series of SolpLq X ker trV is non-trivial abelian ideal of L contained in ker trV . But this contradicts (2˝). So SolpLq X ker trV “ 0. Togetherwith (1˝) this gives (a).

(b) We have rSolpLq, Ls ď SolpLq X ker trV “ 0. So SolpLq ď ZpLq and SolpLq is abelian. Thus (2˝)shows that the elements of SolpLq acts as scalars in V . �

1It suffices to assume that charK does not divide dimK V


Theorem 2.1.11 (Lie). Let V be a standard L-module.

rSolpLq, Ls ď SolpLq X L1 ď NilLpVq.

Proof. Let W be a composition factor for L on V . By 2.1.10 SolpLq X L1 ď AnnLpWq and so SolpLq X L1 ďNilLpVq. �

Corollary 2.1.12. Suppose that L is solvable and V is a standard L-module. Then

(a) L1 ď NilLpVq.

(b) If V is a simple L-module, then V is 1-dimensional.

(c) There exists a series of L-submodules 0 “ V0 ă V1 ă . . . ă Vn “ V with dim Vi “ i.

(d) NilLpVq “ tl P L | l acts nilpotently on Vu.

Proof. Since L is solvable, L “ SolpLq.

(a): Note that L1 “ SolpLq X L1 and so by 2.1.11, L1 ď NilLpVq.

(b): Suppose V is a L-module. Then 2.1.10(b), L “ SolpLq acts as scalars as V . Hence each 1-dimensionalK-subspace of V is L-invariant. Since V is simple, this shows that V is 1-dimensional..

(c) Let 0 “ V0 ă V1 ă . . . ă Vn “ V be an L-composition series of V . Then Vi{Vi´1 is a simpleL-module and so by (b), Vi{Vi´1 is 1-dimensional. So (c) holds.

(d) Clearly each elements of NilLpVq acts nilpotently on V . Now let l P L act nilpotently on V . Then lalso acts nilpotently any simple L-section W of L on V . Thus lW ‰ W. By (b) W is 1-dimensional. HencelW “ 0, l P AnnLpWq and so l P NilLpVq. �

Corollary 2.1.13. Suppose L is a standard. Then

(a) rSolpLq, Ls ď SolpLq X L1 ď NilpLq.

(b) If L is solvable then L1 is nilpotent and there exists a series of ideals 0 “ L0 ď L1 ď . . . ď Ln “ L inL with dim Li “ i.

Proof. Apply 2.1.11 and 2.1.12 to V being the adjoint module L. �

2.2 Tensor products and invariant mapsLemma 2.2.1. Let V and W be L-module.

(a) There exists a unique Lie-algebra action of L on V bK W with

lpvb wq “ plvq b w` vb plwq

for all l P L, v P V and w P W.

(b) There exists a unique Lie-algebra action of L on HomKpV,Wq with

pl f qv “ lp f vq ´ f plvq.

for all l P L, f P HomKpV,Wq and v P V.

2.2. TENSOR PRODUCTS AND INVARIANT MAPS 41

(c) There exists a unique action of L on V˚ :“ HomKpV,Kq with

pl f qv “ ´ f plvq.

for all l P L, f P HomKpV,Kq and v P V.

Proof. For (a) and (b) see Homework 2. (c) follows from (b), by viewingK is L-module annihilated by L. �

Definition 2.2.2. Let V, W and Z be L-modules and let B Ď L.

(a) Let f P HomKpV,Wq. We say that f is B-linear if f pbvq “ bp f vq for all v P V and b P B. HomK,BpV,Wqdenotes the set of K- and B-linear functions from V to W.

(b) Let f : V ˆW Ñ Z be K-bilinear. We say that f is B-invariant if

f pbv,wq ` f pv, bwq “ bp f pv,wqq

for all b P B, v P V,w P W.

Lemma 2.2.3. Let V, W and Z be L-modules and let B Ď L.

(a) HomK,BpV,Wq “ AnnHomKpV,WqpBq.

(b) Let f P HomKpV,Wq. Then AnnLp f q of L is the largest subset X of L such that f is X-linear.

(c) Let f : V ˆW Ñ Z be K-bilinear. Let f be the unique K-linear function

f : V bK W Ñ Z with f pvb wq “ f pv,wq

for all v P V,w P W. Then f is B-invariant if and only of f is B-linear. In particular, AnnLp f q is thelargest subset X of L such that f is X-invariant.

(d) Suppose L annihilates Z and let f : V ˆW Ñ Z be K-bilinear. Then f is B-invariant if and only if

f pbv,wq “ ´ f pv, bwq

for all b P B, v P V,w P W.

Proof. (a): Let b P B and f P HomKpV,Wq. Then

b f “ 0ðñ pb f qv “ 0 for all v P V

ðñ bp f vq ´ f pbvq “ 0 for all v P V - by Definition of b f

ðñ bp f vq “ f pbvq for all v P V

ðñ f is b-linear

(b) follows from (a).

(c): Let b P B. By (a), f is b-linear if and only if b f “ 0.


f is b-linear

ðñ f pbuq “ bp f uq for all u P V bK W - by Definition of b-linear

ðñ f`

bpvb wq˘

“ b`

f pvb wq˘

for all v P V,w P W - by Definition of V bK W

ðñ f`

bvb w` vb bw˘

“ b`

f pvb wq˘

for all v P V,w P W - by Definition of bpvb wq

ðñ f pbv,wq ` f pv, bwq “ b`

f pv,wq˘

for all v P V,w P W - by Definition of f

ðñ f is b-invariant

(d): Since L annihilates Z, b`

f pv,wq˘

“ 0 for all b P B, v P V,w P W. So (d) follows from the definitionof B-invariant. �

Definition 2.2.4. Let f : V ˆW Ñ Z be K-bilinear.

(a) For X Ď V define XK f :“ tw P W | f px,wq “ 0 for all x P Xu.. Often we will just write XK for XK f .

(b) For Y Ď W define fKY “ tv P V | f pv, yq “ 0 for all y P Yu.. Often we will write KY for fKY.

(c) f is called non-degenerate, if VK “ 0 and KW “ 0.

(d) Suppose V “ W. Then radV :“ KV X VK.

(e) f is called K-symmetric if V “ W and for all v,w P V:

f pv,wq “ 0 ðñ f pw, vq “ 0.

Lemma 2.2.5. Let f : V ˆ V Ñ Z be a K-bilinear.

(a) Let X Ď V. Then XK is K-subspace of W.

(b) Y Ď W. Then KY is K-subspace of V.

Proof. Since f is K-bilinear, this should be obvious. �

Lemma 2.2.6. Let f : V ˆ V Ñ Z be a K-bilinear.

(a) If f is symmetric or alternating, then f is K-symmetric.

(b) Suppose f is K-symmetric. Then XK “ KX for all X Ď V. In particular, radV “ VK “ KV.

Proof. Obvious. �

Lemma 2.2.7. f : V ˆW Ñ Z a L-invariant and K-bilinear. Let X be a L-invariant subset of V, then XK isL-submodule of W. In, particular, VK is an L-subspace of W.

Proof. Let w P XK, l P L and x P X. Since f is L-invarinat

(1˝) f plx,wq ` f px, lwq “ l`

f px,wq˘

Since X is L-invariant, lx P X and since w P XK,

2.3. A FIRST LOOK AT WEIGHTS 43

(2˝) f plx,wq “ 0

Also f px,wq “ 0 and so

(3˝) l`

f px,wq˘

“ 0

Substituting (2˝) and (3˝) into (1˝) gives f px, lwq “ 0. Hence lw P XK of all w P XK and l P L. So XK isL-invariant. By 2.2.5(a), XK is a K-subspace of V and so XK is an L-submodule of W. �

Lemma 2.2.8. Let U be the universal enveloping algebra of L and view L as an L-module via the adjointaction.

(a) Let V be an L-module. Then the corresponding action Lˆ V Ñ V, pl, vq Ñ lv is L-invariant.

(b) Lˆ L Ñ L, pa, bq Ñ ra, bs is L-invariant.

(c) Lˆ U Ñ U, pa, uq Ñ au is L-invariant. (Here we view U as an L-module via left multiplication.)

(d) L ˆ L Ñ LpUq, pa, bq Ñ ab is L-invariant. (Here we view LpUq as an L-module via the adjointrepresentation.)

Proof. (a) Let a, l P L and v P V . Define f pl, vq :“ lv. Then since f is an Lie-algebra action

f pra, ls, vq ` f pl, avq “ ra, lsv` lpavq “ aplvq ´ lpavq ` lpavq “ aplvq “ a`

f pl, vq˘

.

and so f is L-invariant.

Since L and U are L-modules via the adjoint action and left multiplication, respectively, (b) and (c) arespecial cases of (a).

(d) Let a, b, c in L and define f pb, cq :“ bc. By 1.3.3 ad paq is derivation of U. Hence

f pra, bs, cq ` f pb, ra, csq “ ra, bsc` bra, cs “ ra, bcs “ ra, f pb, cqs.

�

2.3 A first look at weightsDefinition 2.3.1. (a) A weight for L is a Lie-algebra homomorphism λ : L Ñ LpKq.

(b) ΛpLq “ HomLiepL,LpKqq is the set of all weights of L.

(c) Let λ be a weight of L. Then Kλ is the L-module corresponding to the action

Lˆ KÑ K, pl, kq Ñ λplqk

Lemma 2.3.2. (a) Let λ : L Ñ K be K-linear. Then λ is weight of L if and only if L1 ď ker λ.

(b) ΛpLq is a K-subspace of L˚ “ HomKpL,Kq.

(c) ΛpLq – HomKpL{L1,Kq “ pL{L1q˚ as a K-space.


Proof. (a): λ is a weight if and only if λpra, bsq “ rλpaq, λpbqs for all a, b P L. Since K is commutative, thisholds if and only if λpra, bsq “ 0 for all a, b P L and so if and only if L1 ď ker λ.

(b) and (c) follows from (a). �

Lemma 2.3.3. The map λ Ñ rKλs is bijection between ΛpLq and the set of isomorphism classes of 1-dimensional L-modules. The inverse is given by rVs Ñ trV .

Proof. By 2.1.8(d) isomorphic modules have equal trace function. Let V be a 1-dimensional L-module. Thenlv “ trVplqv for all l P L, v P V and so trV is a weight. Hence the function rVs Ñ trV is a well definedfunction from the set of isomorphism classes of 1-dimensional L-modules to ΛpLq. Note also that V – KtrV

and trKλ “ λ. So the two function in the lemma are inverse to each other. �

Corollary 2.3.4. Let L be standard and solvable. Then the map λ Ñ rKλs is bijection between the weightsof L and isomorphism classes of finite dimensional simple L-modules.

Proof. By 2.1.12(b) the simple finite dimensional L-modules are exactly the 1-dimensional L-modules.Hence the lemma follows from 2.3.3. �

Definition 2.3.5. Let V be an L-module, λ a weight for L and n P NY t8, ˚u.

(a) Vλ :“ VrKλs and Vλpnq :“ VrKλspnq.

(b) λ is called a weight for L on V if Vλ ‰ 0.

(c) ΛVpLq is the set of weights for L on V.

Lemma 2.3.6. Let V be an L-module and λ a weight for L. Then

Vλ “ tv P V | lv “ λplqv for all l P Lu.

Proof. Put W “ tv P V | lv “ λplqv for all l P Lu. Note that W is a K-subspace of V . Let 0 ‰ v P V . Thenv P W if and only if Kv is a L-submodule of V isomorphic to Kλ. By definition, Vλ “ VrKλs and so Vλ is thesum of all L-submodules isomorphic to Kλ. It follows that Vλ “ xWy “ W. �

Lemma 2.3.7. Suppose that L “ Kl for some l P L. Let V be an L-module. λ a weight for L and k :“ λplq.Then

(a) Vλ “ tv P V | lv “ kvu “ AnnVpk ´ lq is the eigenspace for l on V with respect to k. Recall here from1.5.16 that we identified K and L with their images in U and so l´ k P U.

(b) Let n P N. Then Vλpnq “ AnnV`

pl´ kqn˘

.

(c) Vλp8q “Ť

nPN AnnV`

pl´ kqn˘

is the generalized eigenspace of l on V with respect to k.

Proof. (a): Let v P V with lv “ kv and let a P L. Since L “ Ka, a “ tl for some t P K. Hence

av “ ptlqv “ tplvq “ tpkvq “ ptkqv “ ptpλlqqv “ λptlqv “ λpavq

Thus

AnnVpk ´ lq “ tv P V | lv “ kvu “ tv P V | av “ λpaqv for all a P Lu “ Vλ,

where the last equality follows from 2.3.6

2.3. A FIRST LOOK AT WEIGHTS 45

(b): For n “ 0 both sides are 0. Suppose inductively that Vλpnq “ AnnV`

pl ´ kqn˘

. By definition ofVλpn` 1q “ VrKλspn` 1q:

Vλpn` 1q{Vλpnq “`

V{Vλpnq˘

λ

(a)“ AnnV{Vλpnqpl´ kq.

Let v P V . It follows that v P Vλpn` 1q if and only if pl´ kqv P Vλpnq. By the induction assumption thisholds iff pl´ kqnpl´ kqv “ 0, iff pl´ kqn`1v “ 0 and iff v P AnnV

`

pk ´ lqn`1˘

. So (b) holds.

(c): By definition Vλp8q “Ť

nPN Vλpnq and so by (b) Vλp8q “Ť

nPN AnnV`

pl´ kqn˘

. �

Lemma 2.3.8. Let f : V ˆ W Ñ Z be an L-invariant, K-bilinear function of L-modules. Let λ and µ beweights of L.

(a) f`

Vλ ˆWµ

˘

Ď Zλ`µ.

(b) Let i, j P NY t8u. Thenf`

Vλpiq ˆWµp jq˘

Ď Zλ`µpi` j´ 1q,

where Zλ`µp´1q :“ 0 and i` j´ 1 :“ 8 if i “ 8 or j “ 8.

Proof. (a) Let l P L,v P Vλ and w P Wµ.

l f pv,wq “ f plv,wq ` f pv, lwq ´ f is L´ invariant“ f pλplqv,wq ` f pv, µplqwq ´ 2.3.6“ λplq f pv,wq ` µplq f pv,wq

“ pλplq ` µplqq f pv,wq

“ pλ` µqplq f pv,wq.

Thus by 2.3.6 f pv,wq P Zλ`µ.

(b): Suppose first that i P N and j P N. Since Vλp0q “ 0, the lemma holds for i “ 0. So we mayassume that i ą 0 and by symmetry also that j ą 0.

By induction on i` j we also may assume that

(1˝) f pVλpi´ 1q ˆWµp jqq ď Zλ`µpi` j´ 2q and f pVλpiq ˆWµp j´ 1qq ď Zλ`µpi` j´ 2q.

Put

X :“ Vλpiq{Vλpi´ 1q, Y :“ Wµp jq{Wµp j´ 1q, Z :“ Z{Zλ`µpi` j´ 2q.

and define

f : X ˆ Y Ñ Z,`

v` Vλpi´ 1q,w`Wµp j´ 1q˘

Ñ f pv,wq for v P Vλpiq,w P Wµp jq

Note that by (1˝) this is a well-defined function. Since f is K-bilinear and L-invarinat, so is f .Note that by definition of Vλpiq, X “ Xλ. Also Y “ Yµ. Thus by (a)

f`

X ˆ Y˘

Ď Zλ`µ.

Taking inverse images in V,W and Z we see that


f`

Vλpiq ˆWµp jq˘

Ď Zλ`µpi` j´ 1q.

So (b) holds for i, j P N. Taking unions shows that (b) also holds for i “ 8 or j “ 8.�

Corollary 2.3.9. Let V be an L modules, A ď L, λ and µ weights for A and i, j P NY t8u

(a) LλpiqVµp jq Ď Vλ`µpi` j´ 1q

(b) rLλpiq, Lµp jqs Ď Lλ`µpi` j´ 1q.

Proof. (a): By 2.2.8 the map f : L ˆ V Ñ V, pl, vq Ñ v is L-invariant so also A-invariant. Hence using2.3.8:

LλpiqVµp jq “ f`

Lλpiq ˆ Vµp jq˘

Ď Vλ`µpi` j´ 1q.

(b): Just applied (a) with V “ L being the adjoint module for L. �

2.4 Minimal non-solvable Lie algebrasProposition 2.4.1. Let L be a standard Lie algebra such that all proper subalgebras are solvable but L is notsolvable. Then

(a) L “ L1.

(b) SolpLq is the largest ideal of L.

(c) L{SolpLq is simple.

(d) SolpLq “ AnnLpWq, where W is any standard, simple L-module with LW ‰ 0.

(e) SolpLq “ NilLpVq, where V is any standard L-module with LV ‰ 0.

(f) SolpLq “ NilpLq.

Proof. (a) If L1 ‰ L, then both L1 and L{L1 are solvable. Thus L is solvable, a contradiction.

(b) Let I be any proper ideal in L. Then I is solvable and so I ď SolpLq.

(c) By (b), L{SolpLq has no proper ideals.

(d). Since LW ‰ 0, AnnLpWq ‰ L and so by (b), AnnLpWq ď SolpLq. By (a), SolpLq ď L1 and so by2.1.10, SolpLq “ SolpLq X L1 ď AnnLpWq.

(e) Since LV ‰ 0, AnnLpVq ‰ L and so by (b), AnnLpWq ď SolpLq. By 1.7.10, NilLpVq acts nilpotentlyon V and so by 1.6.7 NilLpVq{AnnLpVq is nilpotent. Thus 2.1.4(b) shows that NilLpVq{AnnLpVq is solvable.Hence by 2.1.3(a) NilLpVq is solvable and so NilLpVq ď SolpLq.

By (d) SolpLq ď AnnLpWq for all standard simple L-modules W. By definition,

NilLpVq “č

tAnnLpWq | W is a simple L-section of Vu

and so SolpLq ď NilpVq.

(f) Apply (e) to V “ L. �

2.4. MINIMAL NON-SOLVABLE LIE ALGEBRAS 47

Lemma 2.4.2. Let A be an abelian subalgebra of L with dim L{A “ 1. Then L is solvable and one of thefollowing holds:

(1) A is an ideal in L and L1 ď A.

(2) ZpLq ď A, dim L{ZpLq “ 2, dim L1 “ 1 and L “ A` L1.

Proof. Put B :“ AnnApL{Aq. Since L{A is 1-dimensional, dim A{B ď 1.Note that rB, Ls Ď A and so since A is abelian, rrB, Ls, Ass “ 0. Also rrA, Bs, Ls “ 0 and the Jacobi

Identity show that rrA, Ls, Bs “ 0.Suppose that rA, Ls Ď A. Then A is an ideal in L. Moreover, L{A is 1-dimensional and so abelian. Thus

L1 ď A and (1) holds.Suppose next that rA, Ls Ę A. Since L{A is 1-dimensional, L “ rA, Ls ` A. Since both rA, Ls and A

annihilate B, this gives B ď ZpLq. Since dim L{B “ dim L{A ` dim A{B ď 2 there exit x, y P L withL “ Kx ` Ky ` B. Then L1 “ Krx, ys. Since rA, Ls Ę A we get rx, ys R A. In particular, rx, ys ‰ 0, L1 is1-dimensional, L “ A` L1. From rx, ys ‰ 0 we get ZpLq “ B and dim L{B “ 2. So (2) holds. �

Theorem 2.4.3. Let L be a non-solvable, standard simple Lie-algebra all of whose proper subalgebra aresolvable. Then L – sl2pKq.

Proof. For X ď L define rX :“ NilXpLq. Also let N be the set of elements in L acting nilpotently on L.

1˝. Let X ň L, then X1 ď rX “ X XN . and rX is a nilpotent ideal in X.

Since X ‰ L, X is solvable by assumption. Thus 2.1.11 X1 ď NilXpLq and by 2.1.12(d), rX “ X X N .Since L is non-abelian, L ‰ ZpLq and since L is simple, ZpLq “ 0. Thus L is a faithful L-module and so by1.6.7 rX is nilpotent.

Let 0 ‰ x P l. Since L is finite dimensional, Kl is contained in a maximal subalgebra A of L. Picky P LzA. Then Ky is contained in a maximal subalgebra B of L. Then A ‰ B. Define D :“ AX B.

2˝. L “ A` B and dim L{A “ 1 “ dim A{D “ dim B{D “ dim L{B.

Since L is finite dimensional, there exists simple A-submodule, W{A of L{A. Since A is solvable 2.1.12(b)shows that W{A is 1-dimensional. Let w P WzA. Then rA,Ws Ď W as W is a A-submodule of L. AlsoW “ A ` Kw and since rw,ws “ 0, rw,Ws “ rw, As Ď W. Thus rW,Ws Ď W and so W is a subalgebraof L. The maximality of A implies L “ W. So dim L{A “ 1. By symmetry dim L{B “ 1. Since A ‰ B,L “ A` B. Thus A{D “ A{AX B – A` B{B “ L{B and (2˝) holds.

3˝. rD is an ideal in A.

Note that rA “ NilLpAq is an ideal in A. So if rD “ rA, (3˝) holds. Assume now that that rA ‰ rD. Since rDacts nilpotenly on rA we get from 1.6.12 that rD ň N

rAprDq. So also N

rAprDq Ę rD. By (1˝) applied to X “ A and

D

rAX D “ pAXNq X D “ pAXNq X pDXNq “ rAX rD.

As NrAprDq Ę rD this gives N

rAprDq Ę D. By (2˝) A{D is 1-dimensional and so

A “ NrAprDq ` D ď NAprDq.

Thus (3˝) holds.

4˝. A1 X D “ B1 X D “ rD “ 0.


By (3˝), rD is an ideal in A and by symmetry also in B. Thus rD is an ideal in L “ A ` B, and as L issimple, rD “ 0. By (1˝) A1 X D ď rAX D “ rD. Thus (4˝) holds.

5˝. D is abelian, A is not abelian, A1 is 1-dimensional and A “ A1 ‘ D.

By (4˝) D1 ď A1 X D “ 0. Hence D is abelian. If A is abelian, 2.4.2 shows that L is solvable, acontradiction. Hence A is not abelian and A1 ‰ 0. Since A{D is 1-dimensional and A1 X D “ 0 this showsthat A1 is 1-dimensional and A “ A1 ‘ D.

6˝. L “ A1 ‘ D‘ B1 and trDA1 “ ´trD

B1 .

By (5˝), A “ A1 ‘ D and by symmetry, B “ B1 ‘ D. By (2˝) L “ A` B and so L “ A1 ` D` B1. SinceB1 X A “ B1 X pBX Aq “ B1 X D “ 0 this gives L “ A1 ‘ D‘ B1. Note that A1 is an ideal in A. Thus A1,Dand (by symmetry) B1 are D-submodules of L. It follows that

trDL “ trD

A1 ` trDD ` trD

B1 .

Since D is abelian, trDD “ 0. By 2.1.8(b) trLplq “ 0 for all l P L. Since L “ L1 we conclude trD

L “ 0. ThustrD

A1 “ ´trDB1 and (6˝) holds.

7˝. trDA1 is an isomorphisms. In particular, D is 1-dimensional.

By (6˝), trDA1 “ ´trD

B1 and so ker trDA1 “ ker kerD

B1 . Since A1 is 1-dimensional, ker trDA1 annihilates A1. It

follows that ker trDA1 annihilates, A1 ` D ` B1 “ L. Thus ker trD

A1 ď ZpLq. Since L is not solvable, L ‰ ZpLqand since L is simple we get ZpLq “ 0. Thus ker trA1 “ 0. Hence trD

A1 is 1-1. By (5˝), A is not-abelian, A1 is1-dimensional and A´ A1 ‘ D. Thus D ‰ 0. Since Im trD

A1 ď K, we conclude that trDA1 is an isomorphism.

8˝. There exists a basis pa, b, dq of L with ra, bs “ d, rd, as “ a and rb, ds “ b.

Since trDA1 is onto, we can choose d P D with trA1pdq “ 1. It follows that rd, xs “ x for all x P A1. Also

trB1pdq “ ´trA1 “ ´1 and so rd, ys “ ´y for all y P B1. Since rd, zs “ 0 for all z P D we conclude that A1, Dand B1 are the eigenspaces for d on L with respect the eigenvalues 1,0 and ´1. In particular by 2.3.9 rA1, B1sis contained in the eigenspace with eigenvalue 1` p´1q “ 0. Thus rA, Bs Ď D. If rA1, B1s “ 0, A1 would bean ideal in A` B1 “ L, a contradiction. Let 0 ‰ a P A1 and 0 ‰ b P B1. Then ra, bs “ kd for some non-zerok P K. Replacing b by k´1b we may assume ra, bs “ d. Also ra, ds “ a and rb, ds “ ´rd, bs “ ´p´bq “ b.Since L “ A1 ‘ D‘ B1, pa, b, dq is basis for L and so (8˝) is proved.

From (8˝) we see that L is unique up to isomorphism. By Homework 1, sl2pKq is a simple Lie-algebra overL. It has dimension three and so all proper subalgebras are solvable. Hence sl2pKq fulfills the assumptions ofthe theorem and so L – slpK2q. �

2.5 The simple modules for slpK2q

Definition 2.5.1. Suppose L – slpK2q. A triple px, y, hq is called a Chevalley basis for L if px, y, hq is basisfor L and

rh, xs “ 2x. ry, hs “ 2y and rx, ys “ h.

Note here that if L “ slpK2q, x “ E12, y “ E21 and h “ E11 ´ E22, then px, y, hq is a Chevalley basis forL.

Lemma 2.5.2. Let L – slpK2q with Chevalley basis px, y, hq.

2.5. THE SIMPLE MODULES FOR SLpK2q 49

(a) px, y,´hq is a Chevalley basis for Lop. In particular, the K-linear function

L Ñ L, x Ñ x, y Ñ x, h Ñ ´h

is an anti-automorphism of L.

(b) py, x,´hq is a Chevalley basis for L. In particular, the K-linear function

L Ñ L, x Ñ y, y Ñ x, h Ñ ´h

is an automorphism of L.

(c) py, x,´hq is a Chevalley basis for L. In particular, the K-linear function

L Ñ L, x Ñ y, y Ñ x, h Ñ ´h

is an anti-automorphism of L.

Proof. (a):

rx,´hs “ rh, xs “ 2x, r´h, ys “ ry, hs “ 2y, ry, xs “ ´ry, xs “ ´h.

(b)

r´h, ys “ ry, hs “ 2y r´h, xs “ rx, hs “ 2x and ry, xs “ rx, ys “ ´h.

(c) The function in (c) is the composition of the functions in (a) and (b) and so is an anti-automorphism.�

Lemma 2.5.3. Let L – slpK2q with Chevalley basis px, y, hq and let i P Z`. Then the following holds inU.

(a) hyi “ yiph´ 2iq

(b) xyi “ yix` iyi´1ph´ pi´ 1qq.

(c) yxi “ xyi ´ ixi´1ph` i´ 1q

Proof. Readily verified using the commutator relations and induction on i. See Homework 2 for the details.�

Corollary 2.5.4. Let L – slpK2q with Chevalley basis px, y, hq. Let i P Z` and Suppose charK “ 0. Forl P L define lpiq :“ 1

i! li P U. Then

(a) hypiq “ ypiqph´ 2iq.

(b) xypiq “ ypiqx` ypi´1qph´ pi´ 1qq.

Proof. This follows immediately from 2.5.3 �

Theorem 2.5.5. Suppose K is standard, L “ slpK2q, V is a standard L-module and px, y, hq is the Chevalleybasis for L. Then 0 ‰ v P V and k P K with xv “ 0 and hv “ kv.

Proof. Put A :“ Kx ` Kh. Since rh, xs “ 2x P A, A is solvable subalgebra of L and A1 “ Kx. Let Wbe a simple A-submodule in V . Then by 2.1.12 W is 1-dimensional. In particular, A1 annihilates W. Let0 ‰ v P V0. Then xv “ 0 and hv “ kv for some k P K. �


Theorem 2.5.6. Suppose K ‰ 0, L “ slpK2q, V is a an L-module and px, y, hq is the Chevalley basis for L.Suppose 0 ‰ v P V and k P K with xv “ 0 and hv “ kv. Put m :“ suptn P N | xnv ‰ 0u and let W be thesmallest L-submodule of V containing v.

For i P N put vi :“ yi

i! v “ ypiqv. Then

(a) yvi “ pi` 1qvi`1.

(b) hvi “ pk ´ 2iqvi.

(c) xvi “ pk ´ pi´ 1qqvi´1, where v´1 “ 0.

(d) xyvi “ pi` 1qpk ´ iqvi

(e) pviqmi“0 is a K-basis for W.

(f) If m P N, then m “ k and dim W “ m` 1.

Proof. (a): We compute

yvi “ yypiqv0 “ pi` 1qypi`1qv0 “ pi` 1qvi`1.

and so (a) holds.

(b): Using 2.5.4(a) we have

hvi “ hypiqv0 “ ypiqph´ 2iqv0 “ ypiqpk ´ 2iqv0 “ pk ´ 2iqvi

and so (b) holds.

(c): Using 2.5.4(b)

xvi “ xypiqv0 “ pypiqx` ypi´1qph´ pi´ 1qqv0 “ 0` ypi´1qpk ´ pi´ 1qqv0 “ pk ´ pi´ 1qqvi´1

and so (c) holds.

(d) follows from (a) and (c).(e): Let i P N with i ď m. Then vi ‰ 0. By (a), vi is a eigenvector with eigenvalue k ´ 2i for h. Since

charK “ 0, the k ´ 2i pairwise distinct and so pviqmi“0 is K-linearly independent. Put Z :“ xpviq

mi“0yK. By

(a),(b) and (c), Z is L-invariant. Thus Z is an L-submodule of V containing v “ v0 and so W Ď Z. Notev0 “ v P W and so inductively, vi “

1i vi´1 P W. Thus Z Ď W and Z “ W. Thus pviq

mi“0 spans W and so is a

basis for W.

(f): Suppose that m P N. By (c) with i “ m` 1 we get

0 “ x0 “ xvm`1 “ pk ´ mqvm

As vm ‰ 0 and K is a field, k ´ m “ 0 and so k “ m. As pviqmi“0 is a basis for W, dim W “ m` 1. �

2.6. NON-DEGENERATE BILINEAR FORMS 51

2.6 Non-degenerate Bilinear FormsIn this section we establish some basic facts about non-degenerate bilinear forms that will be used later on.

Lemma 2.6.1. Let V and W be finite dimensional K-spaces and let f : V ˆW Ñ K be non-degenerate andK-bilinear.

(a) There exists a unique K-isomorphism t : W˚ Ñ V, αÑ tα such that αw “ f ptα,wq for all α P W˚,w PW. In particular, dim V “ dim W.

(b) Let pwiqiPI be a basis for W. Then there exists unique basis pviqiPI of V with f pvi,w jq “ δi j for alli, j P I.

(c) Let X be a subspace of V. Then

dim X ` dim XK “ dim V and KpXKq “ X.

In particular, X “ V if and only if XK “ 0.

Proof. Let pwiqiPI be a K-basis for V .

(a) i P I define φi P W˚ by φiw j “ δi j. Since W is finite dimensional, pφiqiPI is a basis for W˚. Inparticular, dim W “ dim W˚. Since f is non-degenerate, the function

r : V Ñ W˚ with prvqw “ f pv,wq.

for all v P V,w P W is 1-1. Hence dim V ď dim W˚ “ dim W. By symmetry dim W ď dim V . Sodim V “ dim W and r is an isomorphism. Putting t :“ r´1. Then for α P W˚ and w P W,

f ptα,wq “`

rptαq˘

w “ αw

(b) For i P I define vi :“ tφi . Then for i, j P I:

f pvi,w jq “ f ptφi ,w jq “ φiw j “ δi j.

(c) Note that the function

g : X ˆW{XK Ñ K, px,w` XKq Ñ f px, yq

is well defined and K-bilinear. Also XKg “ XK{ K“ 0 and pW{XKqKg “ X X WK “ 0. So g is non-degenerate. Thus by (a) dim X “ dim W{XK. Hence

dim X ` dim XK “ dim W “ dim V

By symmetry

dim XK ` dimKpXKq “ dim V

Thus dim X “ dimKpXKq and since X ď KpXKq, X “ KpXKq. Thus (c) holds. �

Definition 2.6.2. A quadratic form on the K-space V is a function q : V Ñ K such that

qpkvq “ k2qpvq


for all k P K, v P V and such that the function

s : V ˆ V Ñ K, pv,wq Ñ qpv` wq ´ qpvq ´ qpwq.

K-bilinear. We call s the symmetric form associated to q. Let u P V with qpuq ‰ 0. Define

u “ qpuq´1u

and

ωu : V Ñ V, v Ñ v´ spv, uqu´

“ v´spv, uqqpuq

u.¯

A function σ : V Ñ V is called an isometry of pV, qq if σ is a K-isomorphism and qpvq “ qpσpvqq for allv P V.

Lemma 2.6.3. Let V be a K-space, let q : V Ñ K be quadratic form with associated bilinear form s, and letu P V with qpuq ‰ 0.

(a) qpv,wq “ qpvq ` spv,wq ` qpwq for all v,w P V.

(b) spv, vq “ 2qpvq for all v P V.

(c) qpuq “ qpuq´1 ‰ 0, ˇu “ u and spu, uq “ 2.

(d) ωuu “ ´u, ωuv “ v for all v P VK and ω2u “ idV .

(e) Let 0 ‰ k P K. Then pkuq “ k´1u and ωku “ ωu. In particular, ωu “ ωu.

(f) ωu is an isometry of q.

(g) Let σ be an isometry of q. Then σpuq “ σpuq and σωuσ´1 “ ωσpuq.

Proof. (a) Follows immediately from the definition of s.

(b) We have qp2vq “ 4qpvq and qpv` vq “ qpvq ` spv, vq ` qpvq.

(c) qpuq “ qpqpuq´1uq “ qpuq´2qpuq “ qpuq´1. So ˇu “ pqpuq´1q´1qpuq´1u “ u. Alsoand spu, uq “ spu,uq

qpuq . So by (b), spu, uq “ 2.(d) ωupuq “ u ´ spu, uqu “ u ´ 2u. If v P uK then also u P VK and spv, uq “ 0. Hence ωupvq “

v´ spv, uqu “ v. Also

ωu`

ωuv˘

“ ωu`

v´ spv, uquq˘

“ ωuv´ spv, uqωuu “ v´ spv, uqu` spv, uqu “ v

(e)

pkuq “ qpkuq´1ku “ k´2qpuq´1ku “ k´1u

andωkupvq “ v´ s

`

v, pkuq˘

ku “ v´ s`

v, k´1u˘

ku “ v´ spv, uqu “ ωupvq.

(f) Since ω2u “ idV , ωu is a K-isomorphism. Also

qpωupvqq “ qpv´ spv, uquq “ qpvq ´ spv, uqspv, uq ` spv, uq2qpuq

“ qpvq ´ spv, uqpspv, uq ´ spv, qpuq´1uqqpvqq “ qpvq.

2.6. NON-DEGENERATE BILINEAR FORMS 53

and so ωu is an isometry.(g)

σpuq “ qpσpuqq´1σpuq “ qpuq´1σpuq “ σpqpuq´1uq “ σpuq

and

pσωuσ´1qpvq “ σ

´

ωu`

σ´1pvq˘

¯

“ σ´

σ´1pvq ´ s`

σ´1pvq, u˘

u¯

“ v´ s`

v, σpuq˘

σpuq

“ v´ s`

v, σpuq˘

σpuq “ ωσpuqpvq.

�

Lemma 2.6.4. Suppose f is a symmetric form on the K-space V and that charK ‰ 2. Then qpvq :“ 12 f pv, vq

is a quadratic form and f is associated symmetric form.

Proof. qpv` wq ´ qpvq ´ qpwq “ 12 p f pv` wq ´ f pv, vq ´ f pw,wq “ f pv,wq �

Lemma 2.6.5. Let V and W be finite dimensional K-spaces and f : V ˆW Ñ K a non-degenerate bilinearform. Define

F : V bK W Ñ pV bK Wq˚ by Fpvb wqpv1 b w1q “ f pv,w1q f pv1,wq

for all v, v1 P V and w,w1 P W.

(a) According to 2.6.1 choose bases pviqiPI and pwiqiPI for V and W such that f pvi,w jq “ δi j. ThenpFpvi b w jqqpi, jqPIˆI is the dual of the basis pv j b wiqpi, jqPIˆI of V bW

(b) F is a K-isomorphism.

(c) Put f ˝ :“ F´1p f q. Then f ˝ “ř

iPI vi b wi. Recall here from 2.2.3(c) that f is the K-linear function

f : V bW Ñ K, with f pvb wq “ f pv,wq.

(d) f p f ˝q “ pdim Vq1K

(e) Suppose that V and W are L-modules and f is L-invariant. Then F is L-linear and L f ˝ “ 0.

Proof. We compute

(1˝) Fpvi b w jqpvl b wkq “ f pvi,wkq f pvl,w jq “ δikδl j “ δpi, jqpl,kq.

Thus (a) holds.

(b) follows directly from (a).

(c) Let t “ř

iPI vi b wi. Then by (1˝)

Fptqpvk b wlq “ÿ

iPI

Fpvi b wiqpvk b wlq “ÿ

iPI

δilδik “ δkl “ f pvk,wlq “ f pvk b wlq

Thus Fptq “ f and so t “ F´1p f q “ f ˝

(d) From (c) we compute


f p f ˝q “ÿ

iPI

f pvi b wiq “ÿ

iPI

f pvi,wiq “ÿ

iPI

1 “ |I| “ dim V.

(e) We have

F`

lpvb wq˘

pv1 b w1q “ Fplvb w` vb lwqpv1 b w1q

“Fplvb wqpv1 b w1q ` Fpvb lwqpv1 b w1q “ f plv,w1q f pv1,wq ` f pv,w1q f pv1, lwq

“ ´ f pv, lw1q f pv1,wq ´ f pv,w1q f plv1,wq “ ´

´

Fpvb wqpv1 b lw1q ` Fpvb wqplv1 b wq¯

“´

´

Fpvb wq`

lv1 b w1 ` v1 b lw1˘

¯

“ ´

´

Fpvb wq`

lpv1 b w1q˘

¯

“l`

Fpvb wq˘

pv1 b w1q˘

Hence F is L-linear. Thus also F´1 is K-linear.Since f is L-invariant, 2.2.3(c) shows that L f “ 0. Since F´1 is K-linear and f ˝ “ F´1p f q this gives

L f ˝ “ 0. �

2.7 The Killing FormDefinition 2.7.1. Let V be finite dimensional L-module V Define

fV : Lˆ L Ñ K, pa, bq Ñ trVpabq.

fV is called the killing form of L with respect to V. In the case of the adjoint module, fL is just called theKilling form of L.

Lemma 2.7.2. Let V be a finite dimension L-module.

(a) fV is a symmetric, L-invariant bilinear form on L.

(b) If I Ĳ L, then IK Ĳ L and rI, IKs ď radp fVq.

(c) rad fV “ VK is an ideal in L.

(d) LetW be the set of factors for some L-series on V. Then

fV “ÿ

WPW

fW .

(e) AnnLpVq ď NilLpVq ď radp fVq.

(f) Suppose L is finite dimensional and let I Ĳ L. Then fI |LˆI “ fL|LˆI .

(g) If L is finite dimensional, then NilpLq ď radp fLq.

Proof. (a): Let a, b P L. By 2.1.8 trVpabq “ trVpbaq and trVpra, bsq “ 0. Hence fV is symmetric, and,since we view K as a L-module annihilated by L, atrVb “ 0 “ trVpra, bsq. Thus trL

V is L-linear. By 2.2.8 thefunction mL : LˆL Ñ U, pa, bq Ñ ab is L-invariant and K-bilinear. Note that fV “ trL

V ˝mL and we concludethat so fV is L-invariant and K-bilinear.

2.7. THE KILLING FORM 55

(b) The first statement follows from 2.2.7. For the second, let i P I, j P IK and l P L. Then r j, ls P IK andso since fV is L-invariant:

fVpri, js, lq “ ´ fVpi, r j, lsq “ 0.

Thus ri, js P radp fVq.

(c): Since rad fV “ LK and L is an ideal in L, (b) shows that rad fV is an ideal in L.

(d): By 2.1.8(c) trV “ř

WPW trW and so (d) holds.

(e): Let W be composition factors for L on V . Then NilLpVqW “ 0 and so also LNilLpVqW “ 0. ThusfWpl, nq “ trWplnq “ 0 for all l P L and n P NilLpVq. Hence (d) shows that fVpl, nq “ 0. So n P radp fVq andthus (e) holds.

(f): By (d), fL “ fI ` fL{I . Observe that I ď AnnLpL{Iq and (e) gives I ď rad fL{I . Thus fI{L|LˆI “ 0and (f) holds.

(g): By 1.7.11(b), NilpLq “ NilLpLq. Thus (g) follows from (e) applied to the adjoint module. �

Theorem 2.7.3 (Cartan’s Solvabilty Criterion). Suppose there exists a standard, faithful L-module with fV “0. Then L is solvable.

Proof. Let L be a counter example with dim L minimal. Then all proper algebras of L are solvable, but Lis not. Thus by 2.4.1(c) L :“ L{SolpLq is simple. Note that L is not solvable and all proper subalgebrasof l are solvable. Thus by 2.4.3, L – slpκ2q. Let px, y, hq be a Chevalley basis for L and x, y, h P L withx “ x` L, y “ yL and H “ h` L. choose rx and ry in L which are mapped onto x and

Since V is a faithful V-module, LV ‰ 0 and so by 2.4.1(e), NilLpVq “ SolLpVq. Let W be the set offactors of an L-composition series on V and let W P W. If LW “ 0, then fWpx, yq “ trWpxyq “ 0. IfLW ‰ 0, then by 2.4.1(e) AnnLpWq “ SolpLq and so W is simple L-module. By 2.5.5 there exist 0 ‰ v P Wand k P K with xv “ 0 and hv “ kv. Since W is simple L-module, W is the L-submodule of W generated byv. Hence 2.5.6 show that dim W “ k` 1 and there exists a K-basis pviq

ki“0 for W with xyvi “ pi` 1qpk´ iqvi

for all 0 ď i ď k. Hence

fWpx, yq “ trWpxyq “kÿ

i“0

pi` 1qpk ´ iq

and so fWpx, yq is a positive integer. Note also that since NilLpVq “ SolpLq ‰ L there exists at least oneW PW with LW ‰ 0. By 2.1.8(c)

fVpx, yq “ÿ

WPW

fWpx, yq “ÿ

WPWLW‰0

fWpx, yq

and so fVpx, yq is a positive integer. But this contradicts fV “ 0. �

Proposition 2.7.4. Let V be standard, faithful L-module. Then

rSolpLq, Ls ď SolpLq X L1 ď NilLpVq ď radp fVq ď SolpLq

In particular, if L is perfect, then SolpLq “ NilLpVq “ radp fVq.


Proof. By Lie’s Theorem 2.1.11

rSolpLq, Ls ď SolpLq X L1 ď NilLpVq.

By 2.7.2(f), NilLpVq ď radp fVq. Finally, Cartan’s Solvabilty Criterion 2.7.3 (applied to radp fVq in place ofL), we have that radp fVq is solvable and so radp fVq ď SolpLq. �

Corollary 2.7.5. Let V be a standard L-module with fV non-degenerate. Then

(a) V is faithful and NilLpVq “ 0.

(b) SolpLq “ ZpLq and SolpLq X L1 “ 0.

(c) If L is solvable, then L is abelian.

Proof. (a) and (b): Since fV is non-degenerate, rad fV “ 0. Using 2.7.4 this gives

rSolpLq, Ls ď SolpLq X L1 ď NilLpVq ď rad fV “ 0

In particular, AnnLpVq ď NilLpVq “ 0 and (a) holds. Moreover, rSolpLq, Ls “ 0 “ SolpLq X L1. HenceSolpLq ď ZpLq ď SolpLq and (b) hold.

(c): If L is solvable, then L “ SolpLq and (b) gives L “ ZpLq. So L is abelian. �

Corollary 2.7.6. Suppose SolpLq “ 0 and V is a standard L-module. Then fV is non-degenerate if and onlyif V is faithful.

Proof. If fV is non-degenerate, then V is faithful by 2.7.5(a). Suppose now that V is faithful. Then by 2.7.4radp fVq ď SolpLq “ 0 and so fV is non-degenerate. �

Lemma 2.7.7. Suppose that L is finite dimensional and fL is non-degenerate. Then SolpLq “ 0.

Proof. By 2.7.2(e), NilpLq ď radp fLq “ 0. So NilpLq “ 0. By 2.1.4(c) this implies SolpLq “ 0. �

Corollary 2.7.8. Suppose L is standard. Then SolpLq “ 0 if and only if fL is non-degenerate.

Proof. If fL is non-degenerate, then by 2.7.7 SolpLq “ 0. If SolpLq “ 0, then also ZpLq “ 0 and so theadjoint module is faithful. So by 2.7.6, fL is non-degenerate. �

Definition 2.7.9. Let f : V ˆ V Ñ Z be K-bilinear and let V1 and V2 be K-subspaces of V. We say that V isthe orthogonal direct sum of V1 and V2 with respect to f and write

V “ V1 k f V1 or V “ V1 k V1

ifV “ V1 ‘ V2 and f pv1, v2q “ 0 “ f pv2, v1q

for all v1, v2 P V2.

Lemma 2.7.10. Let f : V ˆ V Ñ Z be K-bilinear and let V1 and V2 be K-subspaces of V with V “ V1 k V2.Then f is non-degenerate if and only if f |V1ˆV1 and f |V2ˆV2 are non-degenerate.

Proof. Readily verified. �

Proposition 2.7.11. Let V be a finite dimensional L module and suppose that fV is non-degenerate. Let I bean ideal in L with I X SolpLq “ 0. Then

2.7. THE KILLING FORM 57

(a) rI, IKs “ 0.

(b) L “ IË

IK.

(c) IK “ AnnLpIq.

Proof. By 2.7.2(b), rI, IKs ď radp fVq “ 0. So rI, IKs “ 0. Thus (a) holds and

(1˝) IK ď AnnLpIq.

Since I X AnnLpIq is an abelian ideal of L and since SolpLq X I “ 0, we get

(2˝) I X AnnLpIq “ 0.

From (1˝) and (2˝) we have

(3˝) I X IK “ 0.

In particular, dim I ` IK “ dim I ` dim IK. By 2.6.1(c) we have dim I ` dim IK “ dim L and sodim I ` IK “ dim L. Thus I ` IK “ L. Since I X IK “ 0 this gives I “ I ‘ IK and since I and IK areperpendicular to each other, I “ I k IK. So (b) holds.

By (1˝) IK ď AnnLpIq and by (b), L “ I ` IK. Thus using the modular law,

AnnLpIq “ AnnLpIq X L “ AnnLpIq X pI ` IKq “ pAnnLpIq X Iq ` IK (2˝)“ 0` IK “ IK

So (c) holds. �

Theorem 2.7.12. Let V be a finite dimensional L-module and suppose that SolpLq “ 0 and fV is non-degenerate. Then there exists perfect, simple ideals L1, L2, . . . , Ln in L such that

L “ L1 k L2 k . . .k Ln.

Proof. Since fV is non-degenerate,2.7.5 shows that V is a faithful L-module. Since V is finite dimensionalwe conclude that L is finite dimensional and we can induct on dim L.

Suppose first that L is simple. Then L “ L1 or L1 “ 0. Since SolpLq “ 0, this gives L “ L1. So theTheorem holds with n “ 1 and L1 “ L.

So suppose that L is not simple and let I be proper ideal in L. Since SolpLq “ 0 the assumptions of 2.7.11are fulfilled. Hence L “ I

Ë

IK and rI, IKs “ 0. Thus by 2.7.10 f IV and f IK

V are non-degenerate. Also anyideal in I or IK is an ideal in L, SolpIq ď SolpLq “ 0 and SolpIKq ď SolpLq “ 0. By induction we candecompose I and IK into an orthogonal direct sum of perfect simple ideals. Thus the same is true for L. �

Definition 2.7.13. We say that L is a semisimple Lie-algebra if L is the direct sum of simple ideals.

Lemma 2.7.14. (a) All abelian Lie-algebras are semisimple.

(b) L is semisimple Lie-algebra if and only if L is a semisimple L-module via the adjoint action.


Proof. (a): Let pliqiPI be a K-basis for L. Then L “À

iPI Kli and Kli – K is a simple Lie-algebra. Since Lis abelian, each Kli is an ideal in L and L is semisimple.

(b) Suppose L is a semisimple Lie-algebra. Then L “À

iPI Li, where each Li is an ideal in L and Li

is a simple Lie-algebra. Since Li is an ideal, Li is an L-submodule of L. Since Li is simple, Li is a simpleLi-module and so also a simple L-module. Thus L is semisimple L-module.

Suppose L is a semisimple L-module via the adjoint action. Then L “À

iPI Li, there each Li is anL-submodule of L. Then Li is left ideal of L and so also an ideal. Let i, j P I with i ‰ j. Then rLi, L js ď

LiXL j “ 0. Hence L j ď AnnLpLiq and so L “ Li`AnnLpLiq. Since Li is a simple L-module we conclude thatLi is also a simple Li-module. Thus Li is a simple Lie-algebra and so L is semisimple as an Lie-algebra. �

Corollary 2.7.15. Let V be a standard L-module with fV -non-degenerate. Then L “ L1 k ZpLq and L issemisimple

Proof. By 2.7.5 L1 X SolpLq “ 0. So by 2.7.11, L “ L1 k L1K. Since L1K is an ideal in L we get rL1K, Ls ďL1 X L1K “ 0. Thus L1K ď ZpLq. By 2.7.11(c) , AnnLpL1q “ L1K and so ZpLq ď L1K and ZpLq “ L1K.Hence L “ Lk ZpLq. Note that SolpL1q ď L1 X SolpLq “ 0. By 2.7.10 f L1

V is non-degenerate. Hence 2.7.12shows that L1 is direct sum of simple ideals and so semisimple. By 2.7.14(a), ZpLq is semisimple and so alsoL “ L1 k ZpLq is semisimple. �

Corollary 2.7.16. Suppose L is standard and SolpLq “ 0. LetM be the set of minimal ideals of L. Then

(a) fL is non-degenerate.

(b) Let I Ĳ L. Then the ideals of I are exactly the ideals of L contained in I.

(c) Let M PM. Then M is perfect and simple.

(d) L “Ë

M. In particular, L is perfect and semisimple.

(e) Let I Ĳ L and M PM. Then either M ď I or M ď IK.

(f) Let I Ĳ L. Then

I “ë

MPMMďI

M and IK “ë

MPMMęI

M.

Proof. (a) follows from 2.7.8.

(b): By 2.7.11, L “ Ik IK and rI, IKs “ 0. So if J is an ideal of L then rJ, Ls “ rJ, I` IKs “ rJ, Is ď Jand so J is an ideal of L.

(c) Let J Ĳ M. Then by (b), J Ĳ L and so the minimality of M shows that J “ 0 or J “ M. Thus M issimple. Since SolpLq “ 0, M1 ‰ 0 and so M1 “ M.

(d) By 2.7.12 L “ L1kL2k. . .kLn, where each Li is a perfect simple ideal of L. In particular, Li PM. LetM PM. Then M “ M1 ď rM, Ls and so there exists 1 ď i ď n with rM, Lis ‰ 0. Then 0 ‰ rM, Lis ď MXLi

and since M and Li are minimal ideals, M “ M X Li. ThusM “ tL1, . . . Lnu and (d) holds.

(e) Suppose M ę I. Since M is a minimal ideal of L we get rM, Is ď M X I “ 0 and so M ď AnnLpIq.By 2.7.11(c) AnnLpIq “ IK and so M ď IK.

(f) Since SolpIq ď SolpLq “ 0, (d) applied to I in place of L shows that I is the direct sum of itsminimal ideals. Hence (b) implies that I is the direct sum of the minimal ideals of L contained in M. ThusI “

Ë

MPMMďI

M. So also IK “Ë

MPMMďIK

M. Let M PM. By (e), M P IK if and only if M ę I. So (f) holds. �

2.8. NON-SPLIT EXTENSIONS OF MODULES 59

Corollary 2.7.17. Let L be standard. Then SolpLq “ 0 if and only if L is perfect and semisimple.

Proof. If SolpLq “ 0 2.7.16 show that L is perfect and semisimple. Suppose now that L is perfect andL “

Àni“1 Li there each Li is simple ideal of L. Then L “ L1 “

À

L1i and so Li “ L1i . Hence SolpLiq “ 0 andso SolpLq “

Àni“1 SolpLiq “ 0. �

2.8 Non-split Extensions of ModulesDefinition 2.8.1. (a) An extensions of L-modules is a pair of L-modules pW,Vq with W ď V.

(b) An extension of L-modules pW,Vq is called split if there exists a L-submodule X of V with V “ W ‘ X.

(c) Let B and T be L-modules and pW,Vq an extension of L-modules. We say that pW,Vq is an extensionof B by T if W – B and V{W – T as L-modules.

Lemma 2.8.2. An extension pW,Vq of L-modules is split if and only if there exists φ P HomLpV,Wq withφ|W “ idW .

Proof. Suppose first pW,Vq is split. Then V “ W ‘ X for some L-submodule X of V . Let φ be the projectionof V onto X. Then φ is L-linear and φ|W “ idW .

Suppose next that φ : V Ñ W is L-linear with φ|W “ idW . Put X “ ker φ. Then X is L-submodule of V . Ifx P X XW then 0 “ φx “ idW x “ x and so X XW “ 0. Let v P V . Then φv P W and so φpφvq “ φv. Henceφpv´φvq “ 0. and hence v´φpvq P ker φ “ X. Thus v “ φv`pv´φvq P W`X. Thus V “ W`X “ W‘Xand pW,Vq splits. �

Lemma 2.8.3. Let pW,Vq be an extension of L-modules. Let

Φ : HomKpV,Wq Ñ HomKpW,Wq, αÑ α|W

be the restriction map. Then

(a) Φ is L-linear and onto.

(b) kerΦ – HomKpV{W,Wq and kerΦ is submodule of HompV,Wq

(c) S :“ Φ´1pKidWq is an L-submodule of HompV,Wq and S { kerΦ – K as an L-module. (Here we viewK as an L-module annihilated by L.)

(d) pW,Vq is split if and only if pkerΦ, S q is split.

Proof. (a): Clearly Φ is L-linear. Note thate there exists K-subspace X of V with V “ W ‘ X. Letα P HompW,Wq and define

β : V Ñ W, w` x Ñ αw pw P W, x P Xq.

Then Φβ “ α and Φ is onto.

(b): Since Φ is L-linear, kerΦ is an L-submodule of HomKpV,Wq.For α P kerΦ and β : V{W Ñ V, v ` W Ñ α. Then β is well-defined and L-linear. Conversely. for

β P HompV{W,Wq define α : V Ñ W, v Ñ βpv`Wq. Then α P kerΦ. Observe that these to maps are inverseto each other and so (b) holds.


(c): Since idW is L-linear, L annihilates idW P HomKpW,Wq (see 2.2.3(a)). Hence KidW is an L-submodule of HomKpW,Wq. Since Φ is L-linear, S is an L-submodule of HomKpV,Wq. As Φ is onto,ΦpS q “ KidW and so S { kerΦ – KidW – K.

(d) Suppose first that pW,Vq splits. Then by 2.8.2 there exist a L-linear α : V Ñ W with α|W “ idW . SoΦα “ idW . So α P S z kerΦ and since S { ker φ is 1-dimensional, S “ kerΦ ‘ Kα. Since α is L-linear, Lannihilates α and so Kα is an L-submodule of S . Thus pkerΦ, S q splits.

Next suppose that pkerΦ, S q splits and let Y be an L-submodule of S with S “ kerΦ ‘ Y . Then Φ|Y :Y Ñ KidW , α Ñ α|W is an L-isomorphism. Hence Y is a L-module annihilated by L. Hence all α P Y areL-linear. Since Φ|Y is onto there exists α P Y with α|W “ idW . Thus by 2.8.2, pW,Vq splits. �

Lemma 2.8.4. Let pW,Vq be an extension of L-module and X an L-submodule of V with V “ W ` X.Then

(a) X{X XW – V{W.

(b) If pX XW, Xq splits then pW,Vqq splits.

Proof. (a) : V{W “ pX `Wq{W – X{X XW.(b): Suppose pX X W, Xq is split and let Y be an L-submodule of X with X “ pX X Wq ‘ Y . Then

V “ W ` X “ W ` pX XWq ` Y “ W ` Y and W X Y “ pW X Xq X Y “ 0. So pW,Vq is splits. �

Lemma 2.8.5. Let T be L-module and suppose every extension of a simple finite dimensional L-module by Tsplits. Then also every extension of a finite dimensional L-module by T splits.

Proof. Let pW,Vq be an extension of the finite dimensional L-module W by T . If W “ 0, pV,Wq-splits. Sosuppose W ‰ 0 and let B be simple L-submodule of W. Note that V{B

L

W{B – V{W – T . By induction ondim W{B we may assume that pW{B,V{Bq is split. So V{B “ W{B ‘ X{B for some L-submodule X of Vwith B ď X. Then W X X “ B and W ` X “ V and 2.8.4 implies X{B “ X{X XW – V{W – T . Hence Xis an extension of the finite dimensional simple B by T and so splits. Thus by 2.8.4 also pW,Vq-splits. �

Corollary 2.8.6. The following statements are equivalent.

(a) All finite dimensional L-modules are semisimple.

(b) Any extension of a finite dimensional, simple L-module by K splits.

(c) Any extension of finite dimensional L-modules splits.

Proof. (a)ùñ (b): Suppose all finite dimensional L-modules are semisimple and let pW,Vq be an extensionof the finite dimensional L-module by K. Then V is finite dimensional and so semisimple. Then by 1.7.7(a)there exists an L-submodule X of V with V “ W ‘ X. Thus pW,Vq splits.

(b)ùñ (c): Suppose any extension of a finite dimensional, simple L-module byK splits. Then by 2.8.5any extension of a finite dimensional L-module by K splits. Let pV,Wq be an extension of finite dimensionalL-modules and let S and Φ be in 2.8.3. Then S ď HomKpV,Wq and so S is finite dimensional. AlsoS { kerΦ – K and so pkerΦ, S q splits. Thus by 2.8.3 d also pW,Vq splits.

(c)ùñ (a): Suppose any extension of finite dimensional L-modules splits. Let V be a finite dimensionalL-module. If V is simple, V is semisimple. So suppose W is a proper L-submodule of V . By assumptionpW,Vq splits and so V “ W ‘ X for an L-submodule X of V . By induction both W and X are semisimple andso also V is semisimple. �

2.9. CASIMIR ELEMENTS AND WEYL’S THEOREM 61

2.9 Casimir Elements and Weyl’s TheoremIn this section will show that a standard module for a perfect, semisimple Lie algebra is semisimple.

Proposition 2.9.1. Suppose L is finite dimensional and f : L ˆ L Ñ K is a non-degenerate, L-invariant,K-bilinear form. Define Ψ : L bK L Ñ U by Ψpa b bq “ ab. Let f ˝ P L bK L be as in 2.6.5 and putc f “ Ψp f ˝q.

(a) c f P ZpUq X xL2y.

(b) Let pviqiPI and pwiqiPI be bases of L with f pvi,w jq “ δi j. Then

c f “ÿ

iPI

viwi

Proof. View L and U as L-module via the adjoint action. By 2.2.8 the function Lˆ L Ñ U, pa, bq Ñ ab is L-invariant and so by 2.2.3(c), Ψ is L-linear. By 2.6.5(e) L f ˝ “ 0 and so, since c f “ Ψp f ˝q, rL, c f s “ 0. SinceAnnUpc f q is an subalgebra of U and U is generated by L as an associative algebra, this implies rU, c f s “ 0.Thus c f P ZpUq.

By 2.6.5(c) f ˝ “ř

iPI vi b wi and so

c f “ Ψp f ˝q “ Ψ`

ÿ

iPI

vi b wi˘

“ÿ

iPI

Ψpvi b wiq “ÿ

iPI

viwi P xL2y.

�

The elements c f from the preceeding proposition is called the Casimir element of f .

Lemma 2.9.2. Let V be a finite dimensional L-module and suppose that fV is non-degenerate. Define cV “

c fV .

(a) trVpcVq “ dim L1K.

(b) Suppose K is algebraicly closed and V is simple. Then cV acts as a scalar k P K on V. Moreover, oneof the following holds:

(1) charK-dim V and k “ dim Ldim V 1K.

(2) charK| dim V and char K| dim L.

Proof. (a) Let Ψ be defined as in 2.9.1. Recall that f V : LbK L Ñ K is defined by f Vpab bq “ fVpa, bq forall a, b P L. Thus

f Vpab bq “ fVpa, bq “ trVpabq “ trV`

Ψpab bq˘

and so f V “ trV ˝ Ψ. HencetrVpcVq “ trVpΨp f ˝V qq “ f Vp f ˝V q

By 2.6.5(d) f Vp f ˝V q “ dim L and so (a) holds.

(b) Since cV P ZpUq, Schur’s Lemma 1.7.23 applied to the image of cV in EndLpVq shows that cV acts asa scalar k P K. Thus trVpcVq “ dim Vk and (a) gives

dim Vk “ dim L1K.

If charK-dim V , then dim V1K ‰ 0 and (b:1) holds.If charK| dim V , then dim L1K “ dim Vk “ 0K and so charK| dim L and (b:2) holds. �


Theorem 2.9.3 (Weyl). Let L be standard, prefect and semisimple and V a finite dimensional L-module.Then V is semisimple.

Proof. By 2.8.6 its suffices to show that any finite dimensional L-module extension pW,Vq with W simpleand V{W – K splits. Since L{AnnLpVq is also perfect and semisimple we may assume that V is faithful. By2.7.6 fV is non-degenerate. So by 2.9.1 c :“ cV P ZpUq X xL2y. Since V{W – K we have LV Ď W. Hencealso L2V Ď W and cV Ď W. Hence c annihilates V{W and trV{Wpcq “ 0. Since W is simple, Schur’s lemma1.7.23 applied to the image of c in EndLpVq shows that c acts as a scalar k on W. Hence

trVpcq “ trWpcq ` trV{Wpcq “ k dim W1K ` 0 “ k dim W1K.

By 2.9.2(a) trVpcq “ dim L1K and since charK “ 0 we conclude trVpcq ‰ 0. Hence also k ‰ 0. SincecV Ď W and u P ZpUq, we obtain a L-linear function

φ : V Ñ W, v Ñ k´1cv

and since c acts a scalar multiplication by k on W, φ|W “ idW . Thus by 2.8.2 , pW,Vq splits. �

2.10 Cartan Subalgebras and Cartan Decomposition

Definition 2.10.1. H ď L is called selfnormalizing if H “ NLpHq. A Cartan subalgebra of L is a nilpotent,selfnormalizing subalgebra of L.

Lemma 2.10.2. Suppose that L is finite dimensional and |K| ě dim L. Then L has a Cartan subalgebra.

Proof. Choose d P L with H :“ LKd0 p8q minimal. Note that a simple module Kd with weight 0 is the same

as 1-dimensional Kd module annihilates by Kd. So by 1.7.17 H is the largest Kd-submodule of L on whichd acts nilpotently.

In particular, d P H. By 2.3.9

rLKd0 p8q, L

Kd0 p8qs ď LKd

0`0p8q

Thus rH,Hs Ď H and so H is a subalgebra of L. Put V “ L{H. Then V is an H-module and AnnVpdq “VKd

0 “ 0.In particular, the image d: of d in EndKpVq is invertible. Also NLpHq{H ď AnnVpdq “ 0 and so

H “ NLpHq.It remains to show that H is nilpotent. Suppose not and let D ď H such that d P D and D is maximal

with respect to acting nilpotently on H. Then D ‰ H and by 1.6.12 D ň NHpDq. Hence we can chooseh P NHpDqzD. Since the number of eigenvalues for h:d:´1 on V is at most dim V and since |K| ě dim L ądim V , there exists k P K such that k is not an eigenvalue of h:d:´1. Then h:d

:´1 ´ kidV is invertible and soalso h:´kd: invertible. Put l “ h´kd. Then l P NHpDqzD and AnnVplq “ 0. Hence also VKl

0 “ AnnVplq “ 0and VKL

0 p8q “ 0. Note that

LKL0 p8q ` H{H ď VKL

0 p8q

and we conclude LKL0 p8q ď H. The minimality of H “ LKd

0 p8q implies that H “ LKl0 p8q. Thus l acts

nilpotently on H. From 1.6.8 we conclude that D ` Kl acts nilpotently on H, contradicting the maximalchoice of D. �

2.11. PERFECT SEMSISIMPLE STANDARD LIE ALGEBRAS 63

Lemma 2.10.3. Let V be a standard L-module and N a nilpotent subideal in L. Recall that ΛpNq is the setof weights of N and ΛN

V “ tλ P ΛpNq | Vλ ‰ 0u. Then

V “à

λPΛVpNq

Vλp8q

Moreover, each Vpλ8q, λ P ΛVpNq is an L-submodule of V.

Proof. Let λ P ΛVpNq. Since N ĲĲ L, 1.7.21 shows that Vλp8q is an L-submodules of V . So it remains toprove the first statement. If V is the direct sum of two proper N-submodules, we may by induction assumethat the lemma holds for both summands. But then it also holds for V . So we may and do assume

1˝. V is not the direct sum of two proper N-submodules.

Let l P N. Since N is nilpotent 1.6.12 implies that Kl is subideal in N. The Jordan Canonical Form of lshows that V is the direct sum of the generalized eigenspaces of l. By 2.3.7(c) each generalized eigenspace ofl is of the form Vµp8q for some weight µ of Kl. Since KL ĲĲ N, 1.7.21 shows that Vµ is an N-submodule ofV . (1˝) now implies that l has a unique eigenvalue λl on V and so V “ Vµlp8q, where µl is the unique weightof Kl with µlplq “ λl.

Let W be any simple section of N on V . Since N is nilpotent, N is solvable and so by 2.3.4 W – Kλ forsome weight λ of N. In particular, W is also a simple Kl section of V and W – Kµl . Thus λplq “ µlplq “ λl.As l P N was arbitrary, λ is independent of the choice of W. It follows that V is rKλs-module for N and soV “ Vλp˚q “ Vλp8q. �

2.11 Perfect semsisimple standard Lie algebrasIn this section we will investigate the structure of the perfect semisimple standard Lie algebras. For this wefix the following

Notation 2.11.1. (a) L is a perfect, semisimple, standard Lie algebra.

(b) H is a Cartan subalgebra of L.

(c) Λ “ ΛLpHq is the set of weights for H on L.

(d) Φ “ Λzt0u. The non-zero weights for H on L are called the roots of H.

(e) f “ fL, and K is always meant with respect to f .

Lemma 2.11.2. (a) L “À

αPΛ Lαp8q

(b) H “ L0p8q.

(c) rLαp8q, Lβp8qs ď Lα`βp8q for all α, β P ΛpHq.

Proof. (a) This follows from 2.10.3 applied with pL,H,Hq in place of pV, L,Nq.

(b) By 1.7.17 L0p8q is the largest H-submodule of L on which H acts nilpotently. Since H is nilpotentwe get H ď L0p8. If H ň L0p8q the nilpotent action of H on L0p8q implies H ň AnnL0pH,Hq (see 1.6.11),a contradiction to NLpHq “ H.

(c) This follows from 2.3.9. �

Lemma 2.11.3. Let α, β P Λ and h P H.


(a) f is non-degenerate.

(b) trLcαphq “ αphq dim Lc

α.

(c) If αphq ‰ 0, then rh, Lcαs “ Lc

α.

(d) If β ‰ ´α, Lcα K Lc

β.

(e) f |H is non-degenerate.

(f) H is abelian.

(g) ´α P Λ.

(h) Lα ‰ 0.

Proof. (a) : 2.7.16(a).

(b) Let W be the set of factors of a composition series for H on Lαp8q and W P W. Since Lαp8qis a rKαs-module for H, W – Ka. So W is 1-dimensional and h acts by the scalar αphq on W. Thus|W| “ dim Lαp8q and trWphq “ αh. Thus

trLαp8qphq “ÿ

WPW

trWpαq “ αphq dim Lαp8q

(c) Since αphq ‰ 0, W “ rh,Ws for all W PW and so also rh, Lαp8qs “ Lαp8q.

(d) Note that f : Lˆ L Ñ K is L-invariant. Hence by 2.3.8

f`

Lαp8q ˆ Lβp8q˘

ď Kα`βp8q.

Since L annihilates K and α` β ‰ 0, Kα`βp8q “ 0. So f`

Lαp8q ˆ Lβp8q˘

“ 0 and Lαp8q K Lβp8q.

(e) Put X “ř

αPΦ Lαp8q. By 2.11.2 L “ L0p8q ‘ X and H “ L0p8q. By (d), L0p8q K X and soL “ H k X. Since f is non-generate, we conclude from 2.7.10 that f |H is non-degenerate. Thus (e) holds.

(f) Since H is nilpotent, H is solvable. By (e), f HL “ f | H is non-degenerate and so 2.7.5(c) shows that

H is abelian.

(h) Follows from the definition of Λ “ ΛLpHq.

(g) Suppose ´α R Λ. Then by (d) Lβp8q K Lαp8q for all b P 8 and since L “À

βPΛ Lβp8q we getL K Lαp8q. Since f is non-degenerate, this gives Lαp8q “ 0, contrary to (h). �

Notation 2.11.4. Put H˚ “ HomKpH,Kq. Since H is abelian, H˚ “ ΛpHq and so Λ Ď H˚. Since f |H isnondegenerate 2.6.1 yields a K-isomorphism

t : H˚ Ñ H, αÑ tα with αphq “ f ptα, hq.

for all α P H˚, h P H. For α, β P H˚ and h P H define

qphq “12

f ph, hq, f ˚pα, βq “ f ptα, tβq, and q˚pαq “ qptαq “12

f ˚pα, αq

By 2.6.4, q is a quadratic form on H with associated symmetric form f . So also q˚ is quadratic form onH˚ with associated symmetric form f˚. Recall from 2.6.2 that for h P H with qphq ‰ 0 and α P H˚ withq˚paq ‰ 0:

h “ qphq´1h “2

f ph, hqh and ωa : H˚ Ñ H˚, βÑ β´ f ˚pβ, αqα

Lemma 2.11.5. Let α P Φ, x P Lαp8q and y P L´α. Put h “ rx, ys. Then

(a) h P H.


(b) Let β P Λ. Then there exists d P Q with βphq “ dαphq.

(c) h “ rx, ys “ f px, yqtα.

(d) h “ 0 if and only if αphq “ 0 and if only if x K y.

Proof. (a): By 2.11.2 rLαp8q, L´ap8qs Ď Lα`p´aqp8q “ L0p8q “ H.

(b) Put V :“ř

nPZ Lcβ`nα. By 2.11.2(c),

rL˘αp8q, Lβ`nαp8qs Ď L˘a`pβ`nαqp8q “ Lβ`pn˘1qαp8q

and so V is invariant under x and y. Since h “ rx, ys we conclude that trVphq “ 0 and so

0 “ trVphq “ÿ

nPZ

trLβ`nαp8qh “ÿ

nPZ

pβphq ` nαphqq dim Lβ`nαp8q

and soβphq

ÿ

nPZ

dim Lβ`nαp8q “ ´αphqÿ

nPZ

n dim Lβ`nαp8q.

Since β P Λ, Lβ`0αp8q ‰ 0 and soř

nPZ dim Lβ`nαp8q is a positive integer. Multiplying with its inversegives (b).

(c) Let a P H. Since y P L´α, ra, ys “ ´αpaqy. Since f is L-invariant we obtain:

f ph, aq “ f prx, ys, aqq “ ´ f pry, xs, aq “ f px, ry, asq “ ´ f px, ra, ys “ ´ f px,´αpaqyq “ αpaq f px, yq

On the otherhand,

f`

f px, yqtα, a˘

“ f px, yq f ptα, aq “ f px, yqαpaq “ αpaq f px, yq

Since f |H is non-degenerate, this implies h “ f px, yqtα.

(d) Suppose αphq “ 0, then by (b), βphq “ 0 for all β P Λ. It follows that h annihilates is simple sectionof H on L and so h P NilHpLq. Since f |H is non-degenerate, 2.7.5(b) implies NilHpLq “ 0. So h “ 0. Henceh “ 0 if and only if αphq “ 0.

By (c), h “ f px, yqta and so h “ 0 if and only if f px, yq “ 0, that is x K y. �

Lemma 2.11.6. Let a P Φ.

(a) Lα “ Lαp8q is 1-dimensional.

(b) Let n P Z. Then nα P Φ if and only if n “ ˘1.

(c) rLα, L´αs “ Ktα.

(d) f ptα, taq ‰ 0.

Proof. Pick 0 ‰ y P L´α. By 2.11.2(a), L “À

βPΛ Lβp8q and by 2.11.3(d), Lβp8q K L´αp8q for allα ‰ b P Λ. Thus L “ yK ` Lαp8q. Since f is non-degenerate, LK “ 0 and so yK ‰ L. Then there existsx P Lαp8q with x M y. Put h “ rx, ys. By 2.11.5(d) h ‰ 0 and αphq ‰ 0. Put

V :“ Ky‘ H ‘à

nPZ`

Lnαp8q.


By 2.11.2(c), Lαp8q, Lmαp8q Ď Lpm`1qαp8q and so V is invariant under x Since y P L´α, ry,Hs “´rH, ys Ď Ky. Also ry, ys “ 0, L´αp8q, Lmαp8q Ď Lpn´1qαp8q fro all n P Z`. Since L0p8q “ H, thisshows that V is also y-invariant. Since h “ rx, ys we conclude that trVphq “ 0 and so

0 “ trVphq “ ´αphq ` 0`ÿ

nPZ`

nαphq dim Lnαp8q.

Since αphq ‰ 0 we can divide by αphq to obtain:ÿ

nPZ`

n dim Lnαp8q “ 1

Thus dim Lcnα “ 0 for n ą 1 and dim Lαp8q “ 1. So (a) holds. Also (b) holds for positive n. Applying

this result to ´α we see that (b) holds. As Lα and L´α are 1-dimensional, Lα “ Kx and L´α “ Ky. ThusrLα,α , L´αs “ K2rx, ys “ K2h. Since K is algebraically closed K “ K2. By 2.11.5(c) ], h “ f px, yqtα andsince x M y, f px, yq ‰ 0, rLα,α , L´αs “ Ktα and (c) is proved.

Finally 0 ‰ αphq “ αp f px, yqtαq “ f px, yqαptαq “ f px, yq f ptα, tαq and so also (d) holds. �

Lemma 2.11.7. Let α P Φ. Define Hα “ Ktα, Xα “ Lα ` L´α ` Hα and hα “ tα “ 2f ptα,tαq

tα. Then Xα is asubalgebra of L, Xα – slpK

2q and αphαq “ 2. More precisely, if xα P Lα and xá P Lá with rxα, x´αs “ hα,then pxα, xá, hαq is a Chevalley basis for Xα and there exists an isomorphism form Xα to slpK2q with

xα ÞÑ

»

–

0 1

0 0

fi

fl , x´α ÞÑ

»

–

0 0

1 0

fi

fl and hα Ñ

»

–

1 0

0 ´1

fi

fl

Proof. Note first that by 2.11.6(d), f ptα, tαq ‰ 0, so hα is well defined. By 2.11.6(c), rLα, Lás “ Ktα andwe can choose x˘α as in the lemma. Now

αphαq “ f ptα, hαq “ f ptα,2

f ptα, tαqtαq “ 2.

and so rhα, xαs “ αphαqxα “ 2xα and

rx´α, hαs “ ´rhα, x´αs “ ´p´αphαqqx´α “ 2x´α.

Hence Xα “ Kxα ` Kx´α ` Lhα is a subalgebra of L with Chevalley basis pxα, x´α, hαq and the lemmais proved. �

Notation 2.11.8. Let α P Φ. We define an equivalence relation „α on Λ by β „α γ if β´ γ P Zα. We denotethe set of equivalence classes by Λ{Zα. The equivalence classes for of „α are called α-strings. If β, γ are inthe same α-string we say that β ďα γ if γ ´ β P Nα. For β P Φ let β´ rαβα be the minimal and β` sαβα bethe maximal element (with respect to ďα) in the α-string through β. For an α string ∆ define L∆ “

ř

δP∆ Lδ,

Lemma 2.11.9. Let α P Φ and ∆ an α-string.

(a) L∆ is an Xα-submodule andL “

à

∆PΛ{Zα

L∆.

(b) Let β P ∆ and i P Z with iα` β P ∆. Then Lβìα is the eigenspaces for hα on L∆ corresponding to theeigenvector βphαq ` 2i.


Proof. (a) follows immediately from 2.11.2.(b) Since αphαq “ 2, Lβ`iα is contained in the eigenspace for hα corresponding to βphαq ` 2i. Since the

βphαq`2i, i P Z are pairwise distinct we conclude that Lβ`iα is the eigenspace corresponding to βphαq`2i. �

Lemma 2.11.10. Let α P Φ, β P Φ and ∆ the α-string containing β.

(a) βphαq “ f ˚pβ, αq “ rαβ ´ sαβ P Z.

(b) Suppose α P ∆, then

(a) ∆ “ t´α, 0, αu.

(b) L∆ “ Xα ‘ kerα.

(c) kerα “ AnnHpXαq “ H X HKα .

(c) Suppose that α R ∆.

(a) ∆ “ tβ` iα | ´rαβ ď i ď sαβu

(b) L∆ is a simple Xα-module of dimension |∆| “ rαβ ` sαβ ` 1.

(c) AnnL∆pLαq “ Lβ`sαβα.

(d) If α` β P Φ, then rLα, Lβs “ Lα`β.

Proof. We have

(1˝) f ˚pβ, αq “ f ptβ, taq “ f ptβ, tαq “ f ptβ, hαq “ βphαq.

Suppose first that α P ∆. By 2.11.6(b) ZαX Φ “ tα,´αu and so ∆ “ tα, 0,´αu. Since αphαq “ 2 ‰ 0,H “ kerα‘ Hα and so L∆ “ Xα ‘ kerα.

AnnHpXαq “ AnnHpLαq X AnnHpL´αq X AnnHpHq “ kerαX kerp´αq X H “ kerα.

By definition of tα, f ptα, hq “ αphq and so kerα “ H X tKα “ H X HKα . Thus all parts of (b) are proved.

Also if β “ α, then rαβ “ 2, sαβ “ 0 and αphβq “ 2 “ rαβ ´ sαβ. If β “ ´α, then rαβ “ 0, sαβ “ 2 andαphβq “ ´2 “ rαβ ´ sαβ. So also (a) holds in this case.

(c) For an Xα-module I and k P K let dkpIq be the number of composition factor (in a given compositionseries) for Hα on I on which hα acts by multiplication by k. Note that by the Jordan-Holder Theorem dkpIq isindependent of the choice of composition series. LetW be set of factors of a composition series for Xα onL∆.

(2˝) dkpL∆q “ÿ

WPW

dkpWq.

Let W P W. By 2.5.5 there exists 0 ‰ v P V and l P K with xαv “ lv. For i P N define vi “1i! xi´αv.

Since W is simple L-module, W is the smallest L-submodule of W containing v. By 2.5.6 we conclude that lis maximal in N with vl ‰ 0, dim W “ l` 1, pviq

li“0 is a K-basis for W and

(3˝) xvi “ pl´ pi´ 1qqvi´1 and hαvi “ pl´ 2iqvi


for all i P N, where v´1 “ 0. If follows that

dWpkq “

#

1 if k “ l´ 2i for some 0 ď i ď l0 otherwise

.

It follows that either l is even, dWp0q “ 1 and dWp1q “ 0; or l is odd, dWp0q “ 0 and dWp1q “ 1. In eithercase

dWp0q ` dWp1q “ 1.

It follows that

(4˝) dL∆p0q ` dL∆p1q “ÿ

WPW

dWp0q `ÿ

WPW

dWp1q “ÿ

WPW

1 “ |W|

On the otherhand by 2.11.9(b), Lβìα, i P N with β ` iα P Λ, is the eigenspace of hα on L∆ for theeigenvalue βphαq ` 2i. Since α R ∆, β` iα ‰ 0 and so Lβìα is 1-dimensional.

It follows that

dL∆pkq “

#

1 if βphαq ` 2i for some i P N with β` iα P Λ0 otherwise

.

In particular, ant two eigenvalues for hα on L∆ differ by an even integer and we conclude that eitherdL∆p0q and dL∆p1q are both 0 or one of them is 0 and the other is 1. Hence dL∆p0q ` dL∆p1q P t0, 1u. By (4˝)dL∆p0q ` dL∆p1q “ |W| and we conclude that |W| “ 1. Thus L∆ “ W is a simple Xα-module. By (3˝) theeigenspace of hα on W are

(5˝) Kvi, 0 ď i ď l, with eigenvalue l´ 2i

So the largest eigenvalue is l and smallest l´ 2l “ ´l. On the other hand, the eigenspace are

(6˝) Lβìα, i P Z and β` iα P Λ with eigenvalue βphαq ` 2i.

By definition of rαβ and sαβ we have ´rαβ ď i ď sαβ for all i P Z with β ` iα P Λ and both bounds areobtained. Hence the largest eigenvalue is βphαq ` 2sαβ and the smallest is βphαq ´ 2rαβ. Thus

βphaq ´ 2rαβ “ ´l and βphαq ` 2sαβ “ l

Adding this two equations gives 2βphaq ´ 2rαβ ` 2sαβ “ 0 and so βphaq “ rαβ ´ sαβ.By (5˝) any integer between ´l and l which is congruent to l modulo 2 is an eigenvalue. Comparing with

(6˝) shows that b` iα P Λ for all i P Z with ´rαβ ď i ď sαβ So

∆ “ tb` iα | i P Z,´rαβ ď i ď sαβu

In particular, since each Lβìα is 1-dimensional dim L∆ “ rαβ ` sαβ ` 1.Suppose now that α ` β P Φ. Then both Lβ and Lα`β are eigenspace of hta and the eigenvalues of the

second is the eigenvalues of the first minus 2. Comparing with (4˝) shows that Lβ “ Kvi and Lα`β “ vi´1 forsome 1 ď i ď l. By (4˝) xαvi “ pl´ pi´ 1qqvi´1 and so


rLα, Lβs “ rxα, Lβs “ pl´ pi´ 1qqLα`β.

Since i ď l, l´ pi´ 1q ‰ 0 and since charK “ 0, rLα, Lβs “ Lα`β.Also rLα,Kvis “ Kvi´1 and we conclude that AnnL∆pLαq “ Kv0 “ Lβ´rαβα. So all parts of (c) are proved.

�

Lemma 2.11.11. Let α P Φ, ∆ an α-string in Λ and β P ∆.

(a) Let ∆ “ tβ0, β1, . . . βku with β0 ăα β1 ăα . . . ăα βk. Then ωαpβiq “ βk´i.

(b) ωαpβq P Λ.

Proof. (b) Note that β0 “ β´ rαβα and βk “ β` sαβα. By 2.11.10(a) f ˚pβ, αq “ rαβ ´ sαβ and so

ωαpβ0q “ ωαpβ´rαβαq “ ωαpβq´rαβωαpαq “ β´ f ˚pβ, aqα`rαβα “ β´prαβ´sαβqα`rαβα “ β`sαβα “ βk.

Let 0 ď i ď k and observe that βi “ β0 ` iα. Hence

ωαpβiq “ ωαpβ0 ` iαq “ ωαpβiq ` iωαpαq “ βk ´ iα “ βk´i

(a) follows immediately from (b). �

Lemma 2.11.12. H “ř

αPΦ Hα.

Proof. Let h P H with h K Hα “ 0 for all α P Φ. Then αphq “ 0 for all α P Λ. So rh, Lαs “ 0 andh P ZpLq “ 0. Thus h “ 0. Since f |H is non-degenerate and H is finite dimensional the lemma now followsfrom 2.6.1(c). �

Lemma 2.11.13. (a) f ˚pα, βq P Q for all α, β P Φ.

(b) The restriction of f ˚Q

of f ˚ to xΦyQ is a positive definite symmetric Q-bilinear form on xΦyQ.

(c) Any Q-basis of xΦyQ is a K-basis of H˚.

Proof. Let h P H. Then f ph, hq “ trLph2q. Since h annihilates H and acts as βphq on the 1-dimensional spaceLβ we have

(1˝) f ph, hq “ÿ

βPΦ

βphq2.

Since tα P Hα “ rLa, Las we can apply 2.11.5(b) to h “ tα. So for β P Φ exists qβ P Q with βptαq “qβαptaq. Since αptαq “ f ptα, tαq this gives that

(2˝) qβαptαq “ qβ f ptα, tαq

Plugging (2˝) into (1˝) with h “ tα we obtain

f ptα, taq “ÿ

βPΦ

q2β f ptα, tαq2.

By 2.11.6(d) f ptα, tαqq ‰ 0 and so we can divided by f ptα, tαq and conclude that


f ptα, tαq “

˜

ÿ

βPΦ

q2β

¸´1

P Q

Thus

(3˝) f ˚pβ, αq “ f ptβ, tαq “ βptαq(2˝)“ qβ f ptα, taq P Q,

and so (a) is proved.

Put

HQ :“ xtα |P ΦyQ

By (3˝), f ptβ, tαq P Q and since f is bilinear, we get f ph, hq P Q for all h, h P HQ. In particular,βphq “ f ptβ, hq P Q. Hence (1˝) implies that f ph, hq ě 0. Moreover, f ph, hq “ 0 then f ptb, hq “ βphq “ 0for all β P Φ. By 2.11.12

(4˝) H “ÿ

βPΦ

Hα “ xtβ | b P ΦyK.

and so h P H X HK “ 0. Thus f ph, hq ą 0 for all 0 ‰ h P HQ. Hence f |HQ is positive definite. Sincetpαq “ tα we have

HQ “ xtα | α P ΦyQ “ xtpαq | α P ΦyQ “ t`

xα | α P ΦyQ˘

“ t`

xΦyQ˘

.

By defintion of f ˚, t is an isometry from H˚ to H and we conclude that f ˚Q

is positive definite symmetricbilinear form on xΦyQ. Hence (b) holds.

Let B be a Q-basis for HQ. (4˝) gives xByK “ H and so B contains a K-basis D for H. Let h P HQ withh K D. Then h K xDyK “ H. Thus h “ 0 and HQ X xDyKQ “ 0. In particular, f ˚

Qis non-degenerate and

2.6.1(c) shows that xDyQ “ HQ. Thus B “ D is a basis for H.We proved that any Q basis of HQ is K-basis of H. Applying t´1 gives (c). �

Chapter 3

Rootsystems

3.1 Definition and Rank 2 RootsystemsNotation 3.1.1. Throughout this chapter:

(i) F is a subfield of R.

(ii) E a finite dimensional vector space over F.

(iii) p¨, ¨q is positive definite symmetric F-bilinear form on E.

(iv) E7 :“ Ezt0u.

(v) q : E Ñ F, e Ñ 12 pe, eq is the quadratic form associated to p¨, ¨q.

(vi) For e P E7

e “ qpeq´1e “2epe, eq

and ωe : E Ñ E, a Ñ a´ pa, eqe´

“ a´2pa, eqpe, eq

e¯

Note here that this is well-defined, since p¨, ¨q is positive definite and so pe, eq ą 0.

(vii) Φ is a root system (see Definition 3.1.2 below).

Definition 3.1.2. A subset Φ of E7 is called a root system in E provided :

(i)`

α, β˘

P Z (i.e ωβpαq P α` Zβ) for all a, β P Φ.

(ii) ωαpβq P Φ for all a, β P Φ.

(iii) E “ xΦyF

(iv) FαX Φ “ tα,´αu for all α P Φ.

If (i) to (iii) hold, Φ is called a weak root system. If (i) holds then Φ is called a pre-root system.

Note that by 2.11.11, 2.11.12 and 2.11.13 the set Φ non-zero weights of a perfect semisimple standardLie algebra is a root system in xΦyQ. The purpose of this chapter is determine all the roots system up toismomophism. This will be used in later chapters to complete the classifications of the perfect semisimplestandard Lie algebras.

71

72 CHAPTER 3. ROOTSYSTEMS

Definition 3.1.3. Let ∆ Ď E7 and α, β P E7.

(a) Wp∆q is the subgroup of the isometry group of p¨, ¨q generated by the ωα, α P ∆.

(b) W :“ WpΦq is called the Weyl group of Φ.

(c) ∆ :“Ť

∆Wp∆q “ twpδq | w P Wp∆q, δ P ∆u.

(d) ϑαβ is the angle between α and β, that is the real number θ with 0 ď θ ď 180 and cos θ “ pα,βq?pα,αq

?pβ,βq

.

(e) mαβ “`

α, β˘

pβ, αq.

(f) ∆ :“ tδ | δ P ∆u.

Lemma 3.1.4. Let α, β P E7. Then

(a) cos2 ϑαβ “14 mαβ.

(b) 0 ď mαβ ď 4.

(c) If α M β then pα,αq

pβ,βq“pα,βqpβ,αq

.

(d) Fα “ F b iff mαβ “ 4. In this case α “ 12

`

α, β˘

β.

Proof. (a): We have

cos2 ϑαβ “pα, βqpβ, αq

pα, αqpβ, βq“

14

`

α, β˘

pβ, αq “14

mαβ

and so (a) holds.

(b): Since ´1 ď ϑαβ ď 1 we have 0 ď cos2 ϑαβ ď 1 and so (a) implies (b).

(c):

`

α, β˘

pβ, αq“

2pα,βqpβ,βq

2pβ,αqpα,αq

“pα, αq

pβ, βq

.(d): Note that Fα “ Fβ iff ϑαβ P t0˝, 180˝u and iff cos2 ϑαβ “ 1. By (a) we have cos2 ϑαβ “ mαβ and so

this holds iff mαβ “ 4. Suppose now that Fα “ Fβ. Then α “ kβ for some k P F. Since`

β, β˘

“ 2 we get

`

α, β˘

“`

kβ, β˘

“ k`

β, β˘

“ 2k

Hence k “ 12

`

α, β˘

and (d) holds. �

Lemma 3.1.5. Let tα, βu P E7 with`

α, β˘

P Z and pβ, αq P Z and pα, αq ě pβ, βq. Then one of the followingholds:

3.1. DEFINITION AND RANK 2 ROOTSYSTEMS 73

mαβ

`

α, β˘

pβ, αq cosϑαβ ϑαβpα,αq

pβ,βq

0 0 0 0 90˝ ? α K β

1 1 1 12 60˝ 1

1 ´1 ´1 ´ 12 120˝ 1

1 2 1 1?2

45˝ 2

2 ´2 ´1 ´ 1?2

135˝ 2

3 3 1?

32 30˝ 3

3 ´3 ´1 ´?

32 150˝ 3

4 2 2 1 0˝ 1 α “ β

4 ´2 ´2 ´1 180˝ 1 α “ ´β

4 4 1 1 0˝ 4 α “ 2β

4 ´4 ´1 ´1 180˝ 4 α “ ´2β

In particular, if Fα ‰ Fβ and α M β, then |pβ, αq| “ 1, and if α M β then pα,αq

pβ,βqP t1, 2, 3, 4u.

Proof. By 3.1.4(b) 0 ď mαβ ď 4. By assumption,`

α, β˘

and pβ, αq both are integer, so also mαβ “`

α, β˘

pβ, αq is an integer. Thus mαβ P t0, 1, 2, 3, 4u. If mαβ ‰ 0 then α M β and 3.1.4(c) gives pα,βqpβ,αq

“

pα,αq

pβ,βqě 1. Thus |

`

α, β˘

| ě |pβ, αq|. This give the first three and the last two columns of the table. To computeϑαβ note that by 3.1.4(c), cos2 ϑαβ “

14 mαβ. Also cosϑαβ ą 0 if and only if

`

α, β˘

ą 0. �

Definition 3.1.6. For D Ď E define

min-dpDq “ inftpe´ d, e´ dq | d ‰ e P Du

andmax-dpDq “ suptpe´ d, e´ dq | d ‰ e P Du.

We say that D is discrete if min-dpDq ą 0 and that D is bounded if max-dpDq is finite.

Lemma 3.1.7. Let ∆ be linear indepdendent subset of E. Then x∆yZ is discrete.

Proof. Since Z∆ is closed under subtraction we need to show that inf0‰tPx∆yZ pt, tq ą 0.Let d P ∆ and put Σ “ ∆zd. Note that E “ Fd k dK and so for e P E there exists unique fe P F and

e P dK with

e “ fed ` e.

Put Σ “ te | e P Σu. Since ∆ is linearly independent, so is Σ. By induction on ∆ we conclude that

m :“ inf0‰sPxΣyZ

ps, sq ą 0.

Let 0 ‰ t P x∆yZ. Note that function e Ñ e is linear and d “ 0. Also t “ ndd `ř

ePΣ nee for somend, ne P Z and so t “

ř

ePΣ nee P xΣyZ.


If t ‰ 0, we conclude that pt, tq ě m and so

pt, tq “ f 2t pd, dq ` pt, tq ě m

If t “ 0, then since ∆ is linearly independent, t P Fd X Z∆7 “ Zd and so pt, tq ě pd, dq. In either case

pt, tq ě min m, pd, dq ą 0

and the lemma is proved. �

Lemma 3.1.8. Let D Ď E be discrete and bounded. Put l “ min-dpDq, u “ max-dpDq and n “ dim E. Then

|D| ďˆ

1`c

p2n´ 1qul

˙n

.

In particular, D is finite.

Proof. We may assume without loss that D is finite. Replacing p¨, ¨q be 1l p¨, ¨q we may assume that l “ 1.

Suppose first that E is 1-dimensional. Since D is finite there exists c, d P D with pc´ d, c´ dq “ l. Putb “ c ´ d. Then pb, bq “ l “ 1. Since E is 1-dimensional E “ Fb. Note that pkb, bq “ kpb, bq “ k and soe “ pe, bqb for all e. Let m “ |D| ´ 1 and D “ td0, . . . , dmu. Put ki “ pdi, bq and note that ki P F Ď R anddi “ kib. Choose notation such that ki´1 ď ki for all 1 ď i ď m. Let 0 ď i ă j ď m. Then

pd j ´ di, d j ´ diq “ ppk j ´ kiqb, pk j ´ kiqbq “ pk j ´ kiq2pe, eq “ pk j ´ kiq

2

and sob

pd j ´ di, d j ´ diq “

b

pk j ´ kiq2 “ k j ´ ki

Thus?

u ěa

pd j ´ di, d j ´ diq “ k j ´ ki ě?

l “ 1 and so

?u ě km ´ k0 “

mÿ

i“1

ki ´ ki´1 ě

mÿ

i“1

1 “ m

Hence |D| “ m` 1 ď 1`?

u and the lemma holds in the 1-dimensional case.Let E1 be a 1-dimensional F-subspace of E and put E2 “ EK1 . Then E “ E1 k E2 and for i “ 1, 2 let

πi be the corresponding projection of E onto Ei. Let D1 be a subset of π1pDq with min-dpD1q ě1

2n´1 . Letd, e P D1 with d ‰ e and choose s, e P E with π1pdq “ d and π2peq “ 1. Since E “ E1 k E2,

(1˝) pa, aq “ pπ1paq, π1paqq ` pπ2paq, π2paqq

for all a P E and we conclude that pd ´ e, d ´ eq ď`

d ´ e, d ´ e˘

ď u. Hence max-dpD1q ď u. The1-dimensional case treaded above shows that

(2˝) |D1| ď 1`

d

u1

2n´1

“ 1`b

p2n´ 1qu

In particular, we can choose a maximal such D1. For e P D1 put


Dpeq :“"

d P Dˇ

ˇ

ˇ

ˇ

pπ1pdq ´ e, π1pdq ´ eq ă1

2n´ 1

*

.

We claim that

(3˝) D “ď

ePD1

Dpeq.

Suppose for a contradiction that there exists d P DzŤ

ePD1Dpeq. Then pπ1pdq ´ e, π1pdq ´ eq ě 1

2n´1 forall e P D1. But then D1 Y teu has minimum distance at least 1

2n´1 , a contradiction to the maximal choice ofD1. So (3˝) holds.

Now let g, f P Dpeq with g ‰ f . Then π1pgq and π1p f q have distance at most 12n´1 from e and by the

triangular inequality π1pgq and π1p f q have distance at most 22n´1 . Since g and f have distance at least l “ 1,

(1˝) shows that π1pgq and π1p f q have distance at least 1 ´ 22n´1 “ 2n´ 32n´ 1. Thus π2pgq ‰ π2p f q, and

π2pDpeqq has minimum distance at least 2n´32n´1 and maximum distance at most u. Since dim E2 “ n ´ 1, we

inductively have

(4˝) |Dpeq| “ |π2pDpeqq| ď

¨

˝1`d

`

2pn´ 1q ´ 1˘ u

2n´32n´1

˛

‚

n´1

“

ˆ

1`b

p2n´ 1qu˙n´1

Hence

|D|(2˝)ď

ÿ

ePD1

|Dpeq|(4˝)ď

ÿ

ePD

ˆ

1`b

p2n´ 1qu˙n´1

“ |D1|

ˆ

1`b

p2n´ 1qu˙n´1

(3˝)ď

ˆ

1`b

p2n´ 1qu˙ˆ

1`b

p2n´ 1qu˙n´1

“

ˆ

1`b

p2n´ 1qu˙n

and the lemma is proved. �

Lemma 3.1.9. Let Ψ be pre-root system in E. Then xΨyZ is discrete and Ψ is finite.

Proof. Without loss E “ xΨyF and so Ψ contains a basis ∆ of E. Then also ∆ is a basis for E and accordingto 2.6.1(b) there there exists a basis Σ “ pd˚qdP∆ of E with

`

c˚, d˘

“ δcd for all c, d P ∆. Let ψ P Ψ.Then ψ “

ř

dPD kdd˚ for some kd P F and so kd “`

ψ, d˘

. Since Ψ is a pre-root system,`

ψ, d˘

P Z and soψ P xΣyZ. Hence also xΨyZ ď ZΣ. By 3.1.7, xΣyZ is discrete and so also Ψ and ZΨ are discrete.

Put u “ maxδP∆ pδ, δq and let α P Ψ. Since x∆yF “ E, ∆K “ EK “ 0 and so there exists δ P ∆ withpα, δq ‰ 0. Thus 3.1.5 gives

pα, αq ď 4pδ, δq ď 4u.

Hence Ψ is bounded. So Ψ is discrete and bounded and 3.1.8 shows that Ψ is finite.

Lemma 3.1.10. Let ∆ Ď E7.

(a) Wp∆q “ Wp∆q “ Wp∆q.


(b) ∆ “ ∆.

(c) Let X Ď E. Then X is Wp∆q-invariant if and only if ωαpxq P X for all α P ∆. In this case, wpXq “ Xfor all w P Wp∆q.

(d) ∆ is pre-root system if and only if ∆ is a pre-root system.

Proof. (a): Since ∆ Ď ∆ we have Wp∆q Ď Wp∆q. Let α P ∆. Then by definition of ∆ there exists w P Wp∆qand β P ∆ with α “ wpβq. Hence 2.6.3(g) gives ωα “ wωβw´1 P Wp∆q and so Wp∆q Ď Wp∆q. It followsthat Wp∆q Ď Wp∆q. By 2.6.3(f) we have ωα “ ωa for all α P ∆ and so also Wp∆q “ Wp∆q.

(b): By 2.6.3(d) we have wpαqˇ “ wpαq for all w P Wp∆q and α P ∆. Thus

∆ “ twpαqˇ | w P Wp∆q, α P ∆u “ twpαq | w P Wp∆q, α P ∆u “ twpβq | w P Wp∆q, β P ∆u “ ∆.

(c): If X is Wp∆q-invariant, then wpxq P X for all x P X and so also ωαpxq P X for all α P ∆.So suppose ωapxq P X for all α P ∆. Thus ωαpXq Ď X and so also ωαpωαpXqq Ď ωαpXq. By 2.6.3(c),

ω2α “ idE and so X Ď ωαpXq and ωαpXq “ X. Thus wα is contained in T :“ tw P SympEq | wpXq “ Xu.

The letter is a subgroup of SympEq and we conclude that Wp∆q ď T . Hence wpXq “ X for all w P Wp∆q. Inparticular, X is Wp∆q-invariant and (c) is proved.

(d): Suppose ∆ is a pre-root system. Then`

α, β˘

P Z for all α, β P ∆. By 2.6.3(c), ˇβ “ β and so´

α, ˇβ¯

“ pα, βq “ pβ, αq P Z

and so ∆ is a pre-root system.If ∆ is a pre-root system, then ˇ

∆ “ ∆ is pre-root system. �

Lemma 3.1.11. (a) Φ is a weak root system in E. If Φ is a root system, so is Φ.

(b) Φ is W-invariant and wpΦq “ Φ for all w P W.

(c) W acts faithfully on Φ. In particular, W is finite.

Proof. (a): By 3.1.10(d), since Φ is a pre-root system, so is Φ. Let α, β P Φ. Then

ωαpβq “ ωαpβq “ αpβqˇ P Φ

Since Fα “ Fα,

xΦyF “ xΦyF “ E

and so φ is weak root system in E.Suppose now that Φ is a root system and let α, b P Φ. Then β P Fα iff β “ kα for some 0 ‰ k P F. Since

pkβqˇ “ k´1β this holds iff pkβqˇ “ α and if and only if iff kβ “ α. Since Φ is a root system the latter holdsif and only if k “ ˘1. So Φ is a root system.

(b) By definition of pre-root system, ωαpβq P Φ for all α, β P Φ. So (b) follows from 3.1.10(c).

(c) Let w P W with wpαq “ α for all α P Φ. Since E “ xΦyE we conclude that wpeq “ 1 for all e P Eand so w “ idE . Hence W acts faithfully on Φ and W is isomorphic to a subgroup of SympΦq. By 3.1.9, Φ isfinite. Therefore also SympΦq and W are finite. �

Definition 3.1.12. Let Ψ Ď Φ and R a subring of F (with 1 P R and so Z ď R). Then


(a) Ψ is called a (weak) root subsystem of Φ if Ψ is a (weak) root system in xΨyF.

(b) Ψ is called R-closed if Ψ “ xΨyF X Φ.

(c) ΨR denotes the smallest R-closed subset of Φ containing Ψ. ΨR is called the R-closure of Ψ in Φ.

We often just say “subsystem” for “weak root subsystem”

Lemma 3.1.13. Let Ψ Ď Φ and R a subring of F.

(a) SupposeΦ is a root system in E. ThenΨ a weak root subsystem ofΦ if and only ifΨ is a root subsystemof Φ.

(b) Ψ is a subsystem of Φ iff ωαpβq P Ψ for all α, β P Ψ.

(c) Ψ is a subsystem of Φ iff Ψ is invariant under WpΨq.

(d) Ψ Ď Φ and Ψ is the smallest subsystem of Φ containing Ψ.

(e) Let T be an R-submodule of E. Then T X Φ is an R-closed subsystem of Φ.

(f) ΨR is a subsystem of Φ andΨ Ď ΨR “ xΨyR X Φ Ď xΨyR

Proof. (f) Suppose Φ is a root system in E and Ψ is a weak root subsystem of Φ. Let α P Ψ. Then since Φ isa root system, Fα X Ψ Ď Fα X Φ “ tα,´au. Also ´α “ ωαpaq P Ψ and so Fα X Ψ “ tα, αu. Thus Ψ is aroot subsystem of Φ.

(b) Since Φ is a weak root-system and Ψ Ď Φ,`

α, β˘

P Z for all α, β P Ψ. Hence Ψ is a weak root-systemin xΨyF if and only if ωαpβq P Ψ for all α, β P Ψ.

(c): 3.1.10(c) ωαpβq P Ψ for all α, β P Ψ if and only if Ψ is WpΨq-invariant. Hence (c) follows from (b).

(d) Let Σ be any subsystem of Φ containing Ψ. By (c) Σ is invariant under WpΣq. Since Ψ Ď Σ andWpΨq Ď WpΣq this shows that wpψq P Σ for all w P WpΨq and ψ P Ψ. Thus Ψ Ď Σ. In particular, Ψ Ď Φ.

By definition of Ψ we have Ψ “ twpψq | ψ P Ψ,w P WpΨqu and so Ψ invariant under WpΨq. By 3.1.10WpΨq “ WpxΨyq and so xΨy is invariant under WpxΨyq. Thus by (c), Ψ is a subsystem of Φ.

(e) Let α, β P T X Φ. Then

ωβpαq “ α´`

α, β˘

β P α` Zβ P Rα` Rβ ď T,

and so ωαpβq P T X Φ. Hence by (b), T X Φ is a subsystem of Φ. Also

T X Φ Ď xT X ΦyR X Φ Ď T X Φ.

So T X Φ “ xT X ΦyR X Φ and T X Φ is R-closed.

(f) By (e) applied with T “ xΨyR we have that xΨyR X Φ is an R-closed subsystem of Φ containing Ψ.By definition, ΨR is the smallest R-closed subset of Φ containing Ψ and so

ΨR Ď xΨyR X Φ.

Since Ψ Ď ΨR and ΨR is R-closed:

xΨyR X Φ Ď xΨRy X Φ “ ΨR

and thus ΨR “ xΨyR X Φ. Hence by (e) ΨR is a subsystem of Φ containing Ψ. By (d) Ψ is the smallestsubsystem of Φ containing Ψ, and we conclude that Ψ Ď ΨR. �


Lemma 3.1.14. Let ∆ Ď E7 and put

Σ :“ t0 ‰ σ P x∆y | pδ, σq P Z for all δ P ∆u

(a) Σ is a weak root system in xΣyF.

(b) Suppose ∆ is a pre-root system. Then ∆ is a weak root subsystem of Σ.

(c) Suppose ∆ is linearly independent pre-root system. Then ∆ is a root system in x∆yF.

Proof. (a) Let α, β P Σ. We will first show that

1˝.`

α, β˘

s P Z.

Put

U :“ te P E | pe, σq P Z for all σ P Σu.

Since p¨, ¨q is linear in the first coordinate, U is a Z-submodule of E. By definition of Σ, ∆ Ď U and soalso x∆y Ď U. By definition of Σ, Σ Ď x∆y and so Σ Ď U. Thus (1˝) holds.

2˝. ωαpβq P Σ.

By (1˝), pβ, αq P Z and so ωαpβq “ β´ pβ, αqα P x∆y. Let δ P ∆. Then since ωα is an isometry of order2,

pδ, ωαpβqˇq “`

δ, ωαpβq˘

“`

ωαpδq, ωαpωαpβqq˘

“`

ωαpδq, β˘

“`

δ´ pδ, αqα, β˘

“`

δ, β˘

´ pδ, αq`

α, β˘

.

By definition of Σ,`

δ, β˘

P Z and pδ, αq P Z. By (1˝),`

α, β˘

P Z and so pδ, ωαpβqˇq P Z. Thus ωαpβq P Σ bydefinition of Σ.

Note that (1˝) and (2˝) imply that Σ is a root system in xΣyF. So (a) holds.

(b) Since ∆ is a pre-root system, we have`

α, β˘

P Z for all α, β P ∆. Thus ∆ Ď Σ. So by 3.1.13(c), ∆ is aweak root subsystem of Σ.

(c) By (b), ∆ a weak root system and

∆ Ď Σ Ď x∆yZ.

Let k P F and α P ∆. We need to show kα P ∆ if and only of k “ t˘1u. Since ´α “ ωαpaq P ∆ thebackwards direction is obvious. So suppose kα P ∆. By definition of ∆, δ “ wpβq for some β P ∆. Then

kβ “ w´1pkαq P ∆ Ď x∆yZ.

Thus

kβ “ÿ

δP∆

nδδ

for some nδ P Z. Since ∆ is linearly independent over F this gives k “ nβ P Z (and nδ “ 0 for β ‰ δ P ∆.We proved that

(3˝) kα P ∆ implies k P Z.


By 3.1.10 ∆ “ ∆ and since ∆ is a pre-root system so is ∆. As ∆ is F-linearly independent so is ∆.Hence

1kα “ pkαqˇ P ∆ “ ∆

Hence (3˝) applied to 1k , α and ∆ in place of k, α and ∆ gives 1

k P Z. So k P Z and 1k P Z. Hence k “ ˘1

and (c) is proved. �

Definition 3.1.15. Let α, β P Φ. Then ∆ “ pβ` Fαq XΦ is called the α-string of Φ through β. Define a totalordering ďα on ∆ by γ ďα δ if δ ´ γ “ fα for some f P F with f ě 0. Let β ´ rαβα and β ` sαβα be theminimal and maximal element in ∆ with respect to ďα.

Lemma 3.1.16. Suppose Φ is a root system, α, β P Φ and ∆ is the α-string through β.

(a) If α ‰ ´β and pα, βq ă 0, then α` β P Φ. If α ‰ β and pα, βq ą 0, then α´ β P Φ.

(b) ωα leaves ∆ invariant and reverses the ăα ordering. So if ∆ “ tβ0, β1, . . . βmu with β0 ăα β1 ăα β2 ăα

. . . ăα βm, then ωαpβiq “ βm´i.

(c) pβ, αq “ rαβ ´ sαβ.

(d) If β “ ˘α then ∆ “ t˘αu. Otherwise

∆ “ tβ` iα | ´rαβ ď i ď sαβ, i P Zu

In particular, rαβ and sαβ are integers.

Proof. (a) Suppose α ‰ β and pα, βq ă 0. Since pα, αq ą 0, α ‰ β and since FαXΦ “ tα,´αu we concludethat Fα ‰ Fβ. Without loss pα, αq ě pβ, βq. Then by 3.1.5 pβ, αq “ ´1. Thus β` α “ ωαpβq P Φ.

The second statement follows from the first applied to α and ´β.

(b) Let δ P ∆. Then ωαpδq “ δ`pδ, αqα P β`Fα and so ωαpδq P ∆. If γ P ∆ with γ ď δ, then δ “ γ` fαfor a nonnegative f P F. Thus ωαpδq “ ωαpγq ´ fα and so ωαpδq ď ωαpγq.

(c): By (b) ωαpβ0q “ βm, that is

ωαpβ´ rαβαq “ β` sαβα and β´ pβ, αq ` rαβα “ β` sαβα

Since α ‰ 0 this gives pβ, αq “ rαβ ´ sαβ.

(d) If β P Fα, then since Φ is a root system,

∆ “ pβ` Fαq X Φ “ FαX Φ “ tα,´αu

So suppose β R Fa. Without loss β “ β0. Then rαβ “ 0 and 0 ď f ď sαβ for all f P F with β` fα P Φ.Now let f P F with 0 ď f ď sαβ. We need to show that that

δ :“ β` fα P Φ ðñ f P Z.

By (c) rαβ ´ sαβ “ pβ, αq P Z and since rαβ “ 0 we conclude that sαβ P Z. From ωαpβq “ bm “ β` sαβαwe get

ωαpδq “ ωαpβ` fαq “ β` psαβ ´ f qα.


Note that δ P Φ if and only if ωαpδq and f P Z if and only if sαβ ´ f P Z. Also

pωαpδq, αq “ pδ, ωαpαqq “ pδ,´αq “ ´pδ, αq

So replacing f by sαβ ´ f (and thus δ by ωαpδq) if necessary we may assume that pδ, αq ď 0. Choose i P Nmaximal with i ď f . Note that β “ β` 0α P Φ and so such an i exists. Put

γ :“ β` iα P Φ and k :“ f ´ i

Then

γ ` kα “ β` iα` p f ´ iqα “ β` fα “ δ

If k “ 0 then f “ i P Z and δ “ γ P Φ. So we may assume that k ą 0. We claim that k R Z and δ R Φ.Note that

0 ě pδ, αq “ pγ ` kα, αq “ pγ, αq ` 2k ą pγ, αq.

and so by (a) γ ` α P Φ. Thus β` pi` 1q P Φ maximality of i shows i` 1 ą f and so 0 ă k “ f ´ i ă 1.Thus k R Z and to finish the proof of the claim it remains to show δ R Φ.

Suppose for a contradiction that δ P Φ. Then pδ, αq and pγ, αq are both integers. Since pδ, αq “ pγ, αq`2kwe conclude that 2k P Z and as 0 ă k ă 1 we get k “ 1

2 . Hence

δ “ γ `12α and pδ, γq “ 2`

12pα, γq.

Since pδ, γq and pα, γq are integers this shows that pα, γq is an even integer. Since β R Fα, also γ R Fα.As pα, γq ă 0, we conclude from 3.1.5 that pα, γq “ ´2. Hence

ωγpαq “ α´ pα, γqγ “ α` 2γ “ 2ˆ

12α` γ

˙

“ 2δ

Since both δ and ωγpαq are in Φ, we obtained a contradiction to the definition of a root system. �

Definition 3.1.17. (a) The rank of Φ is the minimal size of subset ∆ of Φ with Φ “ ∆.

(b) Define α, β P Φ to be adjacent if α M β. Φ is called connected if this graph is connected. The connectedcomponents of Φ are the connected components of this graph.

Remark 3.1.18. (a) Φ is disconnected if and only if there exists proper subsets Φ1 and Φ2 of Φ withΦ “ Φ1 Y Φ2 and Φ1 K Φ2. (Note here that Φ1 K Φ2 implies Φ1 X Φ2 “ H.

(b) Let Φ1, . . . ,Φn be the connected components of Φ. Then Φ “Ťn

i“1Φi and

E “në

i“1xΦiyF.

Lemma 3.1.19. Let Φ be connected rank two roots system. Then there exists α, β P Φ such

maβ P t1, 2, 3u,`

α, β˘

“ ´mαβ, pβ, αq “ ´1,pα, αq

pβ, βq“ mαβ, ϑαβ “

`

p7` mαβq15˘˝.

Moreover, Φ is as given in Figure 3.1


Proof. Since Φ has rank 2, Φ “ tα, βu for some α, b P Φ. Interchanging α and β, if necessary, we mayassume that pα, αq ě pβ, βq. Replacing α by ´α, if necessary. we may assume that pα, βq ă 0.

If α K β, then t˘αu Y t˘βu is invariant under ωα and ωβ. Thus Φ “ tα, βu “ t˘αu Y t˘βu and Φ isdisconnected. So α M β.

If Fα “ Fb, then since Φ is a root system. β “ ˘α and so Φ “ tαu has rank 1. Thus Fα ‰ Fb.3.1.5 now shows mαβ P t1, 2, 3u, and that

`

α, β˘

, pβ, αq, pα,αqpβ,βq

and ϑαβ are as claimed.

Φ can now be determined using the following algorithm to compute ∆ for a pre-root system ∆:Put A1 “ ∆ and B1 “ ∆ Suppose inductively that An and Bn have been definedCompute

Bn`1 :“ tωδpβq | β P Bn, δ P ∆uzAn and An`1 :“ An Y Bn`1

Note that An Ď ∆ and since ∆ is finite we can repeat until Bn`1 “ H. Then An is invariant under allωδ, δ P ∆. Hence by 3.1.13(c), An is a subsystem of ∆. By 3.1.13(d) ∆ is the smallest subsystem of ∆containing ∆ and so An “ ∆. �

Definition 3.1.20. Let F be subfield of R and let Φ and Φ‚ be root system over F in E and E‚. respectively Asimilarity of root system ρ : pΦ, Eq Ñ pφ‚, E‚q is an F-linear bijection E Ñ E˝ with ρpΦqq “ Φ‚.

Lemma 3.1.21. Let F be subfield of R and ρ : pΦ, Eq Ñ pΦ,E‚q be a similarity of root systems.

(a)´

ρpαq, ˇρpβq¯

“`

α, β˘

for all α, b P Φ.

(b) Let Φ1, . . . ,Φn be the connected components of Φ and put Φ‚i “ ρpΦiq. Then Φ‚1 , . . . ,Φ‚n are the

connected components of Φ‚,

(c) For each 1 ď i ď n there exists ki P F7

i such that

pρpeq, ρpdqq “ kipe, dq

for all d, e P xΦiyF.

Proof. (a) Let α, β P Φ and i P Z. Since ρ is linear and ρpΦq “ Φ‚ we have

β´ iα P Φ ðñ ρpβ´ iαq P Φ‚ ðñ ρpβq ´ iρpαq P Φ‚.

Hence rαβ “ rρpαqρpβq and sαβ “ sρpαqρpβq. By 3.1.16(c)`

α, β˘

“ rαβ ´ sαβ and so´

ρpαq, ˇρpβq¯

“`

α, β˘

.

(b) By (a) we have α K β if and only if ρpαq K ρpβq and so (b) holds.

(c) To simplify notation we may assume that Φ is connected. For α P Φ define kα “pρpαq,ρpαqq

pα,αq. We will

show that kα “ kβ for all α, β P φ. Since Φ is connected it suffices to consider the case α M β. Then using3.1.4(c) and (b)

pα, αq

pβ, βq“

`

α, β˘

pβ, αq“

´

ρpαq, ˇρpβq¯

´

ρpβq, ˇρpαq¯ “

pρpαq, ρpαqq

pρpβq, ρpβqq.

Multiplying with pρpβq,ρpβqq

pα,αqgives kβ “ ka. Put k :“ kα. Then


Figure 3.1: The connected Rank 2 Root Systems

��

@@@@@

��

��

@@@

@@

r1 r1 ` 2r2r1 ` r2

r2´r2

´r1 ` 2r2 ´r1 ´ r2 ´r1

��

TTTTTT

��

TTTTTT

��

TTTTTT

TTTTTT

��

r1 r1 ` r2

´r2 r2

´r1 ´ r2 ´r1

��

TTTTT

TTTTT

��

"""""""

bbb

bb

bb

bbbbbbb

"""

""

""��

TTTTT

��

TTTTT

TTTTT

��

TTTTT

��

r1

r1 ` r2 r1 ` 2r2

r1 ` 3r2

2r1 ` 3r2

r2´r2

´r1 ´ 3r2´r1 ´ 2r2 ´r1 ´ r2

´r1

´2r1 ´ 3r2

3.2. A BASE FOR ROOT SYSTEMS 83

pρpαq, ρpαqq “ kpα, αq

for all α P Φ. So for all α, β P Φ.

pρpαq, ρpβqq “pρpβq, ρpβqq

2

´

ρpαq, ˇρpβq¯

“ kpβ, βq

2

`

α, β˘

“ pα, βq

Since both of the function pd, eq Ñ pρpeq, ρpeqq and pd, eq Ñ kpd, eq are F-bilinear and since E “ xΦyFthis implies that

pρpdq, ρpeqq “ kpd, eq

for all d, e P E.�

3.2 A base for root systemsDefinition 3.2.1. (a) A subset Π of Φ is called base for Φ provided that Π is an F-basis for E and Φ “

Φ` Y Φ´ thereΦ` :“ xΠyN X Φ and Φ´ “ ´Φ`.

(Here xΠyN “ xΠymonoid “ tř

πPΠ nππ | nπ P Nu).

(b) Let Π be a base for Φ. The elements of Π are called simple roots and the element of Φ` are calledpositive roots. For e “

ř

αPΠ fαα P E define

ht e :“ÿ

αPΠ

fα.

ht e is called the height of e with respect to the base Π.

In this section we will show that Φ has a base.

Lemma 3.2.2. Let V be an finite dimensional vector sapce over an infinite field K and let H a finite set ofproper subspace of V. Then V ‰

Ť

H .

By induction dim V . If dim V “ 1, then H “ H and the lemma holds. Suppose that dim W ě 2 andnote Each H P H lies in a hyperplane H of V . Since K is infinite there exists infintely many hyperplane in V ,indeed the hyperplanes of V correspond to the 1-dimensional K-subspaces of the dual V˚. So we can choosea hyperplane W of V with W ‰ H for all H P H . Then W ‰ W X H for all H and so by induction thereexists w P W with w R W X H for all H P H . Then w R

Ť

H . �

Definition 3.2.3. e P E is called regular with respect to Φ if pα, eq ‰ 0 for all α P Φ.

Lemma 3.2.4. Let S be finite subset of Ezt0u. Then there exists e P E with ps, eq ‰ 0 for all s P S . Inparticular, there exist regular elements in E.

Proof. By 3.2.2 V ‰Ť

sPS αK. �

Lemma 3.2.5. Let S be a finite subset of E and e P E. Suppose that

ps, eq ą 0 and ps, tq ď 0

for all s ‰ t P S . Then S is linearly independent.


Proof. Let fs P F withř

sPS fss “ 0. Put S` :“ ts P S | fs ą 0u and S´ :“ S zS`. Then

u :“ÿ

sPS`fss “ ´

ÿ

sPS´fss.

Let s P S` and t P S´. Then fs ą 0, ft ď 0 and ps, tq ď 0 and so ´ fs ftps, tq ď 0. Hence

0 ď pu, uq “

˜

ÿ

sPS`fss,´

ÿ

sPS´fss

¸

“ÿ

sPS

ÿ

tPS´´ fs ftps, tq ď 0.

If follows that u “ 0 and so

0 “ pu, eq “ÿ

sPS`fsps, eq.

Since fs ą 0 and ps, eq ą 0 for all s P S`, this gives S` “ H and so fs ď 0 for all s P S .Note that

ř

sPS p´ fsqe “ ´ř

sPS fss “ 0. So by symmetry, ´ fs ď 0 for all s P S . Hence fs “ 0 for alls P S and S is linearly independent. �

Proposition 3.2.6 (Existence of Bases). Let e P E be regular with respect to Φ. Put

Φè :“ tα P Φ | pα, eq ą 0u, and Πe :“ Φè zpΦè ` Φ

è q.

Then Πe is a base for Φ and Φè is the set of positive roots of Φ with respect to the base Πe .

Proof.Φé :“ tα P Φ | pα, eq ă 0u, Φ` :“ xΠey X Φ, Φ´ “ ´Φ`

Since e is regular, pα, εq ‰ 0 for all α P Φ and so

Φ “ Φè Y Φé .

If α P Φ, the also ´α P Φ. Also α P Φ` if and only if ´α P Φé . Thus

Φé “ ´Φè .

Let α, β P Πe. Then α “ pα ´ βq ` β. Since β P Φè and α R Φè ` Φè this shows α ´ β PR Φè . By

symmetry in α and β also β ´ α R Φè . Hence α ´ β “ ´pβ ´ αq R Φé and it follows that α ´ b R Φ.3.1.16(a) pα, βq ą 0 implies α´ β P φ and we conclude that pα, βq ď 0. Now 3.2.5 shows that Πe is linearlyindependent.

As Φ is finite also tpα, εq | α P Φu is finite. This allows us to prove statements by induction on pα, εq.Let α P Φè . We will show that α P xΦeyN. If α P Πe, this is obvious. If a R Φe, then α “ β` γ for some

β, γ P Φè . Then

pα, eq “ pβ, eq ` pγ, eq, pβ, eq ą 0, pγ, eq ą 0

Thus also pβ, eq ă pα, eq and pγ, eq ă pα, eq. Hence by induction on pα, εq both β and γ are in xΠeyN and soalso α “ β` γ P xΠeyN.

We proved that Φè Ď xΠeyN and so Φè Ď Φ`. Hence Φé Ď ´Φ` “ Φ´ and so

Φ “ Φè Y Φé “ Φ

` Y Φ´

Since Πe is linearly independent, Φ` X Φ´ “ H and we conclude that

3.3. ELEMENTARY PROPERTIES OF BASE 85

Φ`e “ Φ` and Φ´e “ Φ

´.

Note that Φ` Ď xΠeyN Ď xΠeyF and so Φ “ Φ` Y Φ´ Ď xΠeyF. Since E “ xΦyF this gives E “ xΠFyand so Π is an F-basis for E with Φ “ Φ` Y Φ´. Hence Πe is a basis for Φ. �

3.3 Elementary Properties of Base

Lemma 3.3.1. Let ∆ be a linearly independent subset of Φ and α P ∆. Then there exist unique integersnδ, δ P ∆ with

α “ÿ

nδP∆

nδδ.

Moreover,

α “ÿ

δP∆

pδ, δq

pα, αqnδδ,

and pδ,δq

pα,αqnδ is an integer for each δ P ∆.

Proof. By 3.1.14 ∆ Ď x∆yZ and soα “

ÿ

nδP∆

nδδ

for some integers nδ, δ P ∆. Since ∆ is linearly independent, the nδ, δ P ∆ are unique. We compute

α “2

pα, αqα “

ÿ

δP∆

2pδ, δqpα, αqpδ, δq

ndδ “ÿ

δP∆

pδ, δq

pα, αqnδδ.

By 3.1.10(b) α P ∆ “ ∆ and as above 3.1.14 shows that

α “ÿ

mδP∆

mδδ

for some integers mδ, δ P ∆. Note that ∆ is linearly independent and so the last two displayed equation showthat pδ,δq

pα,αqnδ “ mδ P Z for all δ P ∆. �

Lemma 3.3.2. Let Π be a base for Φ.

(a) Π is a base for Φ and pΦ`qˇ “ Φ`, where Φ` :“ xΠyN X Φ.

(b) Let α, β P Π with α ‰ β. Then α´ β R Φ and pα, βq ď 0.

(c) Let α P Φ. Then htα is an integer, htα is positive if and only if α is positive, and α P Π if and only ifhtα “ 1.

(d) Let α P Π. Then Φ`ztαu is ωα-invariant.

(e) Let β P Φ`zΠ. Then there exists α P Π with pα, βq ą 0. For any such α, both ωαpβq and β´ α are inΦ` and htpωαpβq ď htpβ´ αq “ ht β´ 1.

(f) Π “ Φ`zΦ` ` Φ`.


(g) Let β P Φ`. Then there exists α1, α2, . . . αk P Π such that β “řk

i“1 αi and, for all 1 ď j ď k,ř j

i“1 αi P Φ`.

(h) Let β P Φ`. Then there exists α0, α1, α2, . . . αk P Π such that if, if βi is inductively defined β0 “ α0 andβi “ ωαipβi´1q, then β “ βk and βi P Φ

` for all 0 ď i ď k.

(i) Φ “ Π and W “ WpΠq. In particular, for each β P Φ there exists w P W and α P Π with β “ wpαq.

(j) Put δΠ “ 12

ř

φPΦ` φ. Then, for all α P Π, ωαpδq “ δ´ α and pδΠ, αq “ 1.

Proof. (a) Since Π is an F-basis for E so is Π. Let α P Φ` “ xΠyN X Φ. Then

α “ÿ

δPΠ

nδδ

for some nδ P N. Thus by 3.3.1

α “ÿ

δPΠ

pδ, δq

pα, αqnδδ

and pδ,δq

pα,αqnd P Z. Since nδ ě 0 also pδ,δq

pα,αqnd ě 0. Thus α P xΠyN X Φ “ Φ`. Hence pΦ`qˇ Ď Φ`. By

definition of a base Φ “ Φ` Y´Φ` and so

Φ “ pΦ`qˇY p´Φ`qˇ Ď Φ` Y´Φ`

So Π is a base for Φ. Since Φ` X´Φ` “ H we also conclude that pΦ`qˇ “ Φ`.

(b) Note that neither α ´ β nor ´pα ´ βq “ β ´ α is in xΠyN. So α ´ β R Φ` Y ´Φ` “ Φ. By 3.1.16pα, βq ą 0 implies α´ β P Φ. Hence pα, βq ď 0 and (b) is proved.

(c) Since Φ “ Φ` Y Φ´, α “ř

δPΠ nδα, where nδ P Z and either α P Φ` and nδ ě 0 for all δ P Π orα P Φ´ and nδ ď 0 for all δ P N. As htα “

ř

δPΠ nδ and α ‰ 0 we conclude that htα ą 0 for a P Φ` andhtα ă 0 for α P Φ´. Also htα “ 1 exactly when nβ “ 1 for some β P Π and nδ “ 0 for all β ‰ δ P Π. Notethat this holds if and only if α “ β for some β P Π. Thus (c) is proved.

(d) Let β P Φ`ztαu. Then β “ř

δPΠ nδα for some nδ P N. Hence

ωαpβq “ β´ pβ, αqα “ pnα ´ pβ, αqqα`ÿ

α‰δPΠ

nδδ.

Suppose for a contradiction that ωαpβq P Φ´. Then for α ‰ δ P Φ we gave nδ ď 0 and since nδ P N, nδ “ 0.Thus α “ pnα ´ pβ, αqq P FαX Φq “ tαu, a contradiction.

Hence ωαpbq P Φ`. If ωαpβq “ α, then β “ ωαpωαpβqq “ ωαpαq “ ´α, a contradiction to β P Φ`.Thus ωαpβq ‰ α. It follows that ωαpβq P Φ`ztαu and so Φ`ztαu is ωα-invariant.

(e) Let β P Φ`ztαu. Then β “ř

δPΠ nδα for some nδ P N. Thus

0 ă pβ, βq “ÿ

δPΠ

nδpβ, δq

and since nδ ě 0 we conclude that there exists α P Π with pβ, αq ą 0.Suppose now that α P Π with pβ, αq ą 0. Since pβ, αq is an integer, pβ, αq ě 0. Thus

htpωαpbqq “ htpβ´ pβ, αqαq “ htpβq ´ pβ, αq ď htpβq ´ 1 “ htpβ´ αq ă htpβq.

3.3. ELEMENTARY PROPERTIES OF BASE 87

Since β P Φ`, ht β ě 1 and so htpβ´ αq “ htpβq ´ 1 ě 0.As pβ, αq ą 0 and β ‰ α, 3.1.16(a) gives β´α P Φ. In particular, htpβ´αq ‰ 0 and since htpβ´αq ě 0

we conclude that htpβ´ αq ą 0 and β´ α P Φ`. By (d) ωαβ P Φ` and all parts of (e) are proved.

(f) Let α, β P Φ`. Then htpα` βq “ htpαq ` htpβq ě 1` 1 “ 2 and so α` β R Π. Let δ P Φ`zΠ. Thenby (e) δ´ β P Φ` for some β P Φ and so δ “ pδ´ βq ` β P Φ`Φ`. So (f) holds.

(g) By induction on ht β. If ht β “ 1, then β P Π and (g) holds with k “ 1 and α1 “ β.So suppose ht β ą 1. Then β R Π and we can choose α as in (e). Then β´ α P Φ` and htpβ´ aq ă ht β.

By induction (g) holds for β´ α and so also for β.

(h) By induction on ht β. If ht β “ 1, then β P Π and (h) holds with k “ 0 and α0 “ β. Indeedβ “ α0 “ β0 “ βk and β0 “ β P Φ`.

So suppose ht β ą 1. Then β R Π and we can α as in (e). The ωαpβq P Φ` and htpωαpβq ă ht β. Byinduction (g) holds for ωαpβq and so also for β.

(i) Let β P Φ` and choose α0, α1 . . . , αn and β0, . . . , bn. Then

β0 “ α0

β1 “ ωα1pβ0q “ ωα1pα0q,

β2 “ ωα2pβ1q “ pωα2ωα1qpα0q

...

β “ βn “ ωαnpβn´1q “ pωαn . . . ωα1qpα0q

Since αi P Π for all 0 ď i ď n we have α0 P Π and ωαn . . . ωα1 P WpΠq. Thus β P Π. Also

´β “ pωαn . . . ωα1qp´α0q “ pωαn . . . ωα1ωα0qpα0q.

and thus ´β P Π.We proved that Φ Ď Π. Also Π Ď Φ by 3.1.13(d). Hence Φ “ Π and 3.1.10(a) gives WpΠq “ WpΠq “

WpΦq “ W. Hence (i) holds.

(j) Let a P Π. By (d) Φ`ztαu is ωα invariant and

ωα

¨

˝

ÿ

α‰βPΦ`

β

˛

‚“ÿ

α‰βPΦ`

ωαpβq

Thus

ωαpδΠq “ ωα

¨

˝

12

ÿ

βPΦ`

β

˛

‚ “12ωαpαq `

12ωα

¨

˝

ÿ

α‰βPΦ`

β

˛

‚

“ ´12α`

12

ÿ

α‰βPΦ`

β “

¨

˝

12α`

12

ÿ

α‰βPΦ`

β

˛

‚´ α

“ δΠ ´ α

So ωαpδΠq “ δΠ ´ α. Since ωαpδΠq “ δΠ ´ pδΠ, αqα this implies pδΠ, αq “ 1. �


3.4 Weyl ChambersLemma 3.4.1. Let e P E. Then the following statements are equivalent:

(a) e is regular with respect to Φ and Φè “ Φ`.

(b) e is regular with respect to Φ and Πè “ Π.

(c) pδ, eq ą 0 for all δ P Π.

(d) pδ, eq ą 0 for all δ P Φ`.

(e) pδ, eq ă 0 for all δ P Φ´.

Proof. (a) ùñ (b): : By definition of Πe, Πe “ Φè zΦ

è ` Φ

`

i and by 3.3.2(f), Π “ Φ`zΦ` ` Φ`. So ifΦ“e Φ

` then Π“e Π.

(b) ùñ (c): : I f Πe “ Π, then Π Ď Φè and so by Φ`, pδ, eq ą 0 for all δ P Π.

(d) ùñ (c): Just note that Φ` Ď xΠy7N.

(e)ðñ (d): Just note that Φ´ “ ´Φ`.

(d) ùñ (a): Suppose pδ, eq ą 0 for all δ P Φ`. Then Φ` Ď Φè and Φ´ Ď Φé . Hence alsoΦè X Φ

´ Ď Φè X Φé “ H and so Φè Ď Φ`. This Φè “ Φ

é . �

Definition 3.4.2. Let e P E be regular and α P Φ.

(a) The relation „Φ on the set of regular elements is defined by

d „Φ e ðñ Φè “ Φé

(b) Let e P E be regular. Then Cpeq denotes the equivalence classes of „Φ containing e. The equivalenceclasses of „Φ are called the Weyl chambers of Φ.

(c)Pαpeq :“ td P E | pα, eqpα, dq ą 0u and rPαpeq :“ td P E | pα, eqpα, dq ě 0u

(So Pαpeq consists of all elements of E on the same side of the hyperplane αK )

(d) rCpeq :“Ş

αPΦrPαpeq, rCpeq is called the closed Weyl chamber Φ containing ε.

(e)

CΠ :“ te P E | pδ, eq ą 0 for all δ P Πu and rCΠ :“ te P E | pδ, eq ě 0 for all δ P Πu

The elements of Cπ are called dominant with respect to Π and the elements of Cπ are called strictlydominant.

Lemma 3.4.3. (a) Let e P E. Then e P CΠ if and only if e is regular and Φè “ Φ` and if and only if e isregular and Πe “ Π.

(b) δΠ P CΠ. In particular, CΠ ‰ H.

(c) Cpeq “ CΠ “Ş

αPΦ Pαpeq and rCpeq “ rCΠ for all e P CΠ.

3.4. WEYL CHAMBERS 89

(d) The functionΞ : Cpeq Ñ Πe

is a well-defined bijection from the set of Weyl-chambers of Φ to the set of bases of Φ with well-definedinverse

Θ : ΠÑ CΠ.

Proof. (a) By definition of CΠ, e P CΠ if and only if pe, δq ą 0 for all δ P ∆. Thus (a) follows from 3.4.1.

(b) By 3.3.2(j) pδΠ, αq “ 1 for all α P Π. So pδΠ, αq ą 0 and δΠ P CΠ.

(c) Let d P E. By (a), Φè “ Φ`. By definition of Cpeq, d P Cpeq if and only if Φ`d “ Φè , that isΦ`d “ Φ

`. By (a) this holds if and only if d P CΠ. Hence Cpeq “ CΠ.Also

d P CΠðñ pd, αq ą 0 @α P Π ´ definition of CΠ

ðñ pd, φq ą 0 @φ P Φ` ´ Φ` Ď xΠy7

N

ðñ pd, φqpe, φq ą 0 @φ P Φ` ´ φ P Φ` “ Φè , so pe, φq ą 0

ðñ pd, φqpe, φq ą 0 and pd,´φqpe,´φq ą 0 @φ P Φ` ´ pd, φqpe, φq “ pd,´φqpe,´φq

ðñ pd, φqpe, φq ą 0 @φ P Φ ´ Φ “ Φ` Y´Φ`

ðñ d P Pφpeq @φ P Φ ´ definition of Pφpeq

ðñ d Pč

φPΦ

Pφpeq

So CΠ “Ş

φPΦ Pφpeq. A similar argument, replacing ą by ě, gives rCΠ “Ş

φPΦrPφpeq. By definition,

rCΠpeq “Ş

φPΦrPφpeq and so (c) is proved.

(d): Let d be regular. We apply (a) with Π “ Πd and so Φ` “ Φd. This gives Φè “ Φ`

d if and only ifΠe “ Πd. So d P Ce if and only if Πd “ Πe. So the function Ξ is well-defined. As Πe “ Πe, (a) gives e P CΠe

and so by (b), Ce “ ΠCΠe. So Θ ˝ Ξ is the identity.

By (b) there exists e P CΠ. By (a) Πe “ Π and by (c), CΠ “ Ce. Thus CΠ is a chamber and Θ is welldefined. Moreover,

ΞpΘΠq “ ΞCΠ “ ΞCe “ Πe “ Π

and so also Ξ ˝ Θ is the identity. �

Proposition 3.4.4. Let e P E and d be an element of maximal height in e “ twpeq | w P Wu. Then d isdominant. In particular, there exists w P W with wpeq P rCΠ. If in addition e is regular, then d is regular andd P CΠ.

Proof. Let α P Π. Then ωαpdq P e and ωαpdq “ d ´ pd, αqα has height ht d ´ pd, αq. The maximal choiceof ht d implies that pd, αq ě 0. Thus d P Ppαq and d P C. If e is regular, then since Φ is w-invariant also d isregular. Thus pd, αq ‰ 0 and so pd, αq ą 0 and d P CΠ. �

Definition 3.4.5. Given a base Π of Φ.

(a) Let w P W. Then w is called a simple reflection if w “ ωα for some α P Π.


(b) lpwq is the smallest t P N such that there exists simple reflections σ1, . . . , σt with w “ σtσt´1 . . . σ2σ1.

(c) npwq :“ |wpΦ`q X Φ´|.

(d) Φw :“ tα P Φ` | wpαq P Φ´u.

Remark 3.4.6. Let w P W.

(a) lpwq “ 0 if and only w “ 1.

(b) lpwq “ 1 if and only if w is a simple reflection.

(c) Φw “ Φ` X w´1pΦ´q, wpΦwq “ wpΦ`q X Φ´ and npwq “ |Φw|.

Lemma 3.4.7. Let w P W and σ1, . . . , σt be simple reflections with

w “ σtσt´1 . . . σ1.

Let α P Φ` with wpαq P Φ´. Then there there exists 1 ď s ď t with

wωα “ σtσt´1 . . . σs`1σs´1σs´2 . . . σ1.

In particular, lpwωαq ď t ´ 1.

Proof. For 1 ď i ď t put ρi “ σiσi´1 . . . σ1 and βi “ ρipαq. Note that w “ ρt and βt “ wpαq is negative.So we can choose 1 ď s ď t minimal such that βs is negative. Then βs´1 is positive. Since σs is a simplerefections there exists δ P Π with σs “ ωδ.

βs´1 P Φ` and ωδpβs´1q “ ωspβs´1q “ βs P Φ

´

Thus by 3.3.2(d) Φ`ztδu is ωδ-invariant and so that βs´1 “ δ. Thus

σs “ ωδ “ ωβs´1 “ ωρs´1pαq “ ρs´1ωαρ´1s´1,

σsρs´1 “ ρs´1ωα

ρs “ σsρs´1 “ ρs´1ωα,

ρsωα “ ρs´1

Multiplying the last equation with σtσt´1 . . . σs`1 from the left gives the lemma. �

Lemma 3.4.8. Let w P W and α P Π.

(a) If wpαq is negative, then lpwωαq “ lpwq ´ 1 and npwωαq “ npwq ´ 1

(b) If wpαq is positive, then lpwωαq “ lpwq ` 1 and npwωαq “ npwq ` 1.

(c) lpwq “ npwq.

Proof. (a) Put t :“ lpwq and choose simple roots σ1, . . . , σt with w “ σtσt´1 . . . σ1. Then by 3.4.7

lpwωαq ď t ´ 1 “ lpwq ´ 1.

Since wωαωα “ w we have lpwq ď lpwωαq ` 1 and so lpwωαq “ lpwq ´ 1. Thus the first statement in (a)hold.

3.4. WEYL CHAMBERS 91

Let Σ “ Φ`ztαu. By 3.3.2(d) ωαpΣq “ Σ. Hence also

wpΣq X Φ´ “ pwωαqpΣq X Φ´.

Now wpαq P Φ´ while

pwωαqpαq “ wp´αq “ ´wpαq P Φ`

Thus

wpΦ`q X Φ´ “ wpΣq X Φ´ Y twpαqu and pwωαqpΦ`q X Φ´ “ wpΣq X Φ´.

and so npwωαq “ npwq ` 1.

(b) Note that wωαpαq “ wp´αq is negative. So (b) follows from (a) applied to wωα.

(c) Since lp1q “ 0 “ np1q this follows from (a) and (b) and induction on lpwq �

Theorem 3.4.9. (a) Let w P W and e P rCΠ with wpeq P rCΠ. Then wpeq “ e and w P WpΠ X eKq. If, inaddition, e P CΠ, then w “ 1.

(b) Let D and D1 be Weyl chambers. Then there exists a unique w P W with wpDq “ D1.

(c) Let Π and Π1 be bases for Φ. Then there exists a unqiue w P W with wpΠq “ Π1.

(d) |W| is the number of Weyl chambers.

(e) There exists a unique element w0 P W with npw0q maximal. Moreover npw0q “ lpw0q “ |Φ`|,w0pΠq “ ´Π, w0pΦ

`q “ Φ´ and w20 “ 1.

Proof. (a) We will prove the fist statement by induction on lpwq. If lpwq “ 0, then w “ 1 and it holds. Sosuppose lpwq ą 0 and pick a simple root α with lpwωαq ă lpwq. Then by 3.4.8, wpαq is negative. As both eand wpeq are in rCW we have

0 ď pe, αq “ pwpeq,wpαqq ď 0

Thus pe, αq “ 0 and α P eK. It follows that wωαpeq “ wpeq. By induction on lpwq, pwωaqpeq “ e andwωα P WpΠX eKq. Hence also wpeq “ e and w “ pwωaqωα P WpΠX eKq.

If e P C, then ΠX eK “ H. Thus w P WpHq “ 1 and (a) is proved.

(b) Without loss of generality D1 “ CΠ. Pick d P D. Then by 3.4.4 there exists w P W with wpdq P C.Since W acts on the set of Weyl chambers this gives

wpDq “ wpCpdqq “ Cpeq “ CΠ.

Let w1 be any element of W with w1pDq “ CΠ. Then

pw1w´1qpeq “ w1pdq P CΠ

and so by (a) appled to w1w´1 and e we have w1w´1 “ 1 and so w1 “ w.

(c): By 3.4.3(d) the exists an W-equivariant bijections between the set of Weyl-chambers. So (c) followsfrom (b).

(d): By (c) the function w Ñ wpCπq is a bijection between W and the set of Weyl-chambers.


(e). Note that npwq ď |Φ´| “ |Φ`| for all w P W. Also npwq “ |Φ`| if and only if wpΦ`q “ Φ´ and soif and only if wpΠq “ ´Π.

Note that´Π is a base for Φ with Φ`p´Πq “ Φ´. Thus by (c) such a unique w0 P W with w0pΠq “ ´Π.As w2

0pΠq “ Π we get w20 “ 1. So (e) holds. �

Definition 3.4.10. A subset S of Ezt0u is called acute (obtuse) if ps, tq ě 0 (ps, tq ď 0) for all s ‰ t P E.

Lemma 3.4.11. Let ∆ be a linear independent obtuse preroot system in E. Then ∆ is base for the root system∆ in x∆yF.

Proof. Put Φ :“ ∆. Then by 3.1.14 Φ is a root system. Let α P Φ and write α “ř

βP∆ nββ with nβ P Z. Weneed to show that the non-zero nβ all have the same sign. Suppose not and choose such an α with

ř

βP∆ |nβ|minimal. Since pα, αq is positive there exists δ P ∆ with nδpα, δq ě 0. Replacing α by ´α if necessary wemay assume that nδ ą 0. Then also pα, δq is positive. Note that α R Fδ and so by 3.1.16(a), α ´ δ is a root.Now

α´ δ “ pnδ ´ 1qδ`ÿ

δ‰βP∆

nββ

By the minimal choice of α, the non zero coefficents of α ´ δ are either all positive or all negative. Letδ ‰ β P ∆. If nβ ą 0 then nγ ą 0 for all γ P ∆, contrary to our assumptions. Hence nβ ď 0. Thus alsonδ ´ 1 ď 0. But nδ ´ 1 ě 0 and so nδ ´ 1 “ 0 and nδ “ 1. Since pβ, δq ď 0 this implies that

pα´ δ, δq “ÿ

δ‰βP∆

nβpβ, δq ě 0.

So`

α, δ˘

“`

α´ δ, δ˘

``

δ, δ˘

ě`

δ, δ˘

“ 2. Note also that α ‰ δ and so 3.1.5 implies that pδ, δq ă pα, αq.On the other hand by 3.3.1 pδ,δq

pα,αqnδ is an integer. Since nδ “ 1, this implies pδ, δq ě pα, αq, a contradiction. �

Lemma 3.4.12. Let e P E and suppose that for each w P W there exist nonnegative fδ P F, δ P Π suchwpeq “ ˘

ř

δPΠ fδδ. Then e P Fα for some φ P Φ.

Proof. Suppose this is false. Let α P Φ. Then e R Fα and so αK ę eK. Thus aK X eK ň eK and 3.2.2 showthat

eK ‰ď

αPΦ

aK X eK

Hence there exists u P eK with u R αK for all α P Φ. By 3.4.4 there exists w P W with wpuq P rCΠ. Letd P π. Then pwpuq, δq ě 0. On the other hand w´1pδq P Φ and so u R w´1pδqK. Therefore

pwpuq, δq “`

u,w´1pδq˘

‰ 0 and pwpuq, δq ą 0.

By hypothesis wpeq “ ˘ř

δPΠ fδδ for nonnegative fδ in F. Thus

0 “ pu, eq “ pwpuq,wpeqq “

˜

wpuq,˘ÿ

δPΠ

fδδ

¸

“ ˘ÿ

δPΠ

fdpwpuq, δq

Since fd ě 0 and pwpuq, δq ą 0 for all δ P Π this gives fδ “ 0 for all δ P Π. Hence also wpeq “ 0 and soe “ 0 P Fα for all α P Φ, a contradiction. �

Chapter 4

Uniqueness and Existence

4.1 Simplicity of semisimple Lie algebrasWe say that a subset Ψ of a root system is closed if α` β P Ψ whenever α, β P Ψ with α` β P Φ.

Proposition 4.1.1. Let L be a standard Liealgebra with SolpLq “ 0. Let H be a Cartan Subalgebra andΦ theset of roots for H on L. LetΨ Ď Φ be closed. Put LpΨq “ xLα | α P ΨyLie and HpΨq “ xHα | α P ΨX´ΨyLie.

(a) LpΨq “À

αPΨ Lα ‘ HpΨq. In partcular, if β P Φ, then Lβ P LpΨq if and only if β P Ψ.

(b) Let A ď L with rA,Hs ď A and put ∆ “ tα P Φ | Lα ď Au. Then ∆ is closed and A “ Lp∆q`pAXHq.If, in addition, A is perfect, then A “ Lp∆q.

(c) Let A ď L. Then A is an ideal iff A “ ApΨq for some closed Ψ Ď Φ with Φ “ ΨY pΦX ΨKq.

(d) Let D be the set of connected components of Φ. Then Lp∆q,∆ P D are the simple ideals in L andL “

À

∆PD Lp∆q.

Proof. (a): Let A :“ LpΨqÀ

αPΨ Lα ‘ HpΨq. Let α P Ψ X ´Ψ. Then Hα “ rLα, Las P LpΨq and soA ď LpΨq. So it suffices to show that A is a subalgebra of L. Clearly rHpPsiq, As ď A. Now let α, β P Ψ. Ifα`β P Φ, then sinceΦ is closed, α`β P Ψ and so rLα, Lβs “ Lα`b ď A. If α`b “ 0, then α “ b P ΨX´Ψ,rLα, Lβs “ Hα ď HpΨq ď A. If 0 ‰ α ` β R Φ, then rLα, Lβs “ 0 ď A. Thus A is a subalgebra and (a) isproved.

(b). Let α, β P ∆ with α ` β P Φ. Then Lα`β “ rLα, Lβs ď A and so ∆ is closed. Since A is invariantunder H, A “

À

λPΛ Aλ. For α P Φ, Lα is 1-dimensional so either Lα “ Aα or Aα “ 0. Also A0 “ AXH andso A “

À

αP∆ Lα ‘ pH X Aq “ Lp∆q ` pH X Aq.Since rLpΨq,Hs ď LpΨq and H is abelian we conclude that rA, As ď LpΨq. So if A is perfect, A “ LpΨq.(c) Let Ψ “ tα P Φ | Lα ď A and Σ “ ΦX ΨK.Suppose that A “ LpΨq and Φ “ Ψ X Σ. Note that rLpΨq,Hs ď LpΨq. Let β P Σ and a P Ψ. Then

α ` β is neither perpendicular to Σ nor to Ψ and so α ` β R Φ. Also a ‰ ´β and so rLα, Lβs “ 0. HenceLpΨq, Lβs “ 0 and LpΨq is an ideal.

Suppose next that A is an ideal. By 2.7.16 A is perfect and so by (b), A “ LpΨq. Let α P Φ and supposethat β M α for some β P Ψ. We claim that α P Ψ. If ´α P LΨ, then Lα “ rrLα, L´αs, L´αs ď A and soα P Ψ. So we may assume that pα, βq ă 0 and so by 3.1.16(b), α` β P Φ. So Lα`b “ rLα, Lβs ď LpΨq. Also´β P Ψ and so α “ pα` βq ` p´βq P Ψ. Hence Φ “ ΨY Σ.

93

94 CHAPTER 4. UNIQUENESS AND EXISTENCE

(d) By 2.7.16, L “Àn

i“1 Li, where L1, . . . , Ln are the simple ideals in L. By (b), Li “ LpΨiq, where Ψi isa closed subset of Φ. Moreover, Ψi K Ψ j for all i ‰ j and Φ “

Ťni“1Ψi. Let ∆ be a connected component of

Ψ. Then clearly ∆ Ď Ψi for some i. By (c) Lp∆q is an ideal and so since Li is simple, Lp∆q “ Li “ LpΨiq. (a)implies ∆ “ Ψi. Therefore the Ψi are exactly the connected components of Φ. �

4.2 Generators and relations

Proposition 4.2.1. Let L be a standard Lie algebra with SolpLq “ 0. Let H be a Cartan subalgebra, Φ bethe set of roots for H on L and Π an base for Φ. For α P ˘Π pick 0 ‰ xα P Lα with rxα, x´αs “ hα. Putcαβ “ pβ, αq. Then for all α, β P Π:

(Rel: a) rhα, hβs “ 0.

(Rel: b) rhα, xβs “ cαβxβ and rhα, x´βs “ ćαβx´β.

(Rel: c) rxα, x´βs “ δαβhα.

(Rel: d) If α ‰ β then xćαβ`1α ˚ xβ “ 0 and xćαβ`1

´α ˚ x´β “ 0.

Proof. (Rel: a) holds since H is abelian. (Rel: b) follows since x˘β P L˘b .Note that for β ‰ α, 0 ‰ α ´ β R Φ and so rxα, x´βs P Lα´β “ 0. By choice of the x˘α, rxα, x´αs “ hα

and so (Rel: c) holds.By induction on i we have xi

α ˚ xb P Lβìα. Since α ´ β is not a root, rαβ “ 0. So by 2.11.10(a)cαβ “ pβ, αq ““ rαβ´ sαβ “ ´sαβ. Hence sαβ “ ćαβ and so β`pćαβ` 1qα R Φ. Thus the first statementin (Rel: d) holds. By symmetr y also the second holds. �

Definition 4.2.2. Let V be a vector space and φ P EndpVq. Then φ is called locally nilpotent if for all v P Vthere exists n P N with φnpvq “ 0.

Lemma 4.2.3. Let V be a vectorspace over the field K and φ, ψ locally nilpotent endomorphism. Supposethat charK “ 0.

(a) There exists a unique ρ P EndpVq with ρpvq “řn

i“01i!φ

ipvq whenever v P V and n P N with φn`1pvq “

0. We denote ρ by eφ and byř8

i“0φi

i! .

(b) If rφ, ψs “ 0 then eφ`ψ “ eφeψ.

(c) eφ is invertible with inverse e´ψ.

Proof. Readily verified.

Lemma 4.2.4. Let A be a K algebra, δ a derivation and suppose that charK “ 0. Then

(a)δn

n!pabq “

ÿ

i` j“n

δi

i!paq

δ j

j!pbq

(b) If δ is locally nilpotent then eδ is an automorphism of A.

4.2. GENERATORS AND RELATIONS 95

Proof. (a) This is the usually product rule for differentiation. Indeed for n “ 1, it is just the defintion of aderivation and the general formula can be established by induction on n.

(b)

eδpaqeδpbq “ÿ

i“0

δi

i!paq ¨

8ÿ

j“0

δ j

j!pbq “

8ÿ

n“0

ÿ

i` j“n

δi

i!paq

δ j

j!pbq

“

8ÿ

n“0

δn

n!pabq “ eδpabq �

Lemma 4.2.5. Suppose that K is standard and L – slpK2q with Chevalley basis px, y, hq. Let V be anL-module. Then the following are equivalent:

(a) V is the sum of the finite dimensional L-submodule.

(b) V is the direct sum of finite dimensional simple L-submodules.

(c) x and y act locally nilpotent on V and h is diagonliazible on V.

By 2.9.3, (a) implies (b). Also by 2.5.6, (b) implies (c).Suppose now that (c) holds. Let W be the sum of the finite dimensional L-submodule. Suppose V ‰ W.

Since x is locally nilpotent on V , there exists v P VzW with xv P W. Then there exists a finite dimensionalL-submodule T of W with xv P T . Hence AnnV{T pxq ę V{T and since h is diagonalizible on V , there existseigenvector u ` T P AnnV{T pxq with u R W. Let U{T be the smallest L-submodule of V{T containingu ` T . Since y is locally nilpotent, there exists m P N with ymu “ 0. 2.5.6 now implies that U{T is finitedimensional. But then U is is finite dimensional and so u P U ď W, a contradiction.

Thus V “ W and (c) implies (a). �

Lemma 4.2.6. Let K be standard, S and H subalgebra of V, α P ΛpHq and V an L-module. Supposethat

(i) S – slpK2q with Chevalley basis px, y, hq.

(ii) H is abelian, h P H, x P Lα and y P L´α

(iii) V is the direct sum of the weight spaces for H.

(iv) x and y are locally nilpotent.

Then for each λ P H˚ dim Vλ “ dim Vλ´λphqα.

Proof. Since xα P Lα, rxa,Hs ď Kxa. Similarly, rx´α,Hs ď L´α. Since H is abelian, rhα,Hs “ 0 and asS “ Kxα ` Khα ` Kx´α we conclude that rS ,Hs Ď S . Thus S ` H is Lie-subalgebra of L and we mayassume that L “ S ` H. If the lemma is true for the composition factors for L on V its also true for V . Sowe may assume that V is a simple L-module. Since x is locally nilpotent, AnnVpxq ‰ 0 and so by (iii) thereexists µ P H˚ and 0 ‰ v P Vµ with xv “ 0. Since y is locally nilpotent, there exists m P N minimal withym`1v “ 0. Note that vi :“ yiv is a weightvector for H of weight µ´ iα. By 2.5.6(b) W “ Kvi | 0 ď i ď my isS -invarinat and µphq “ m. Since W is also H invariant, W is an L-submodule and so W “ V . Let λ P ΛVpHq.Then λ “ µ ´ iα for some 0 ď i ď m and Vλ “ Kvi is 1-dimensional. From x P Lα and rh, xs “ 2x wehave αphq “ 2 and so λphq “ m ´ 2i. Thus λ ´ λphqα “ µ ´ iα ´ pm ´ 2iqα “ µ ´ pm ´ iqα and soλ´ λphqα P ΛVpHq. �

Theorem 4.2.7 (Serre). Let Φ be a root system with basis Π and L the Lie-algebra over K generated byelements xα, x´α, hα, α P Π subject to the relations (Rel: a)-(Rel: d) in 4.2.1:


(Rel: a) rhα, hβs “ 0.

(Rel: b) rhα, xβs “ cαβxβ and rhα, x´βs “ ćαβx´β.

(Rel: c) rxα, x´βs “ δαβhα.

(Rel: d) If α ‰ β then xćαβ`1α ˚ xβ “ 0 and xćαβ`1

´α ˚ x´β “ 0.

Suppose that charK “ 0. Then

(a) dim L “ |Φ| ` |Π|.

(b) SolpLq “ 0.

(c) H :“ xhα | α P ΠyLie is a Cartan subalgebra.

(d) There exists a Q-linear map ρ : ΦÑ H˚ with ρpαqphβq “`

α, β˘

for all α P Φ, β P Π.

(e) ρpΦq is the set of roots ΦLpHq for H on L and Φ.

(f) ρ : ΦÑ ΦLpHq is similarity of root systems.

Proof. We will fist only use the first three relations. So let L be the Lie-algebra over K generated by elementsxα, x´α, hα, α P Π subject to the relations (Rel: a)– (Rel: c). . Note that L isomorphic to a quotient of L. Asa first step in determining the structure of L we derive some information about L. Let H “ xhα | α P ΠyLie.Note the relation (Rel: a) ensures that H is abelian and spanned by the hα as a K-space.

To make sure that L ‰ 0 we will construct a non-zero representation of L. Let V be a K-space with basisvpαq, α P Π. Let T be the tensor algebra over V , so T has basis B “ tbn

i“1vpαiq | n P N, αi P Πu. Forb “ bn

i“1vpαiq P B and α P Π define

cαb “

nÿ

i“1

cααi .

In particular ( for the case n “ 0) cα1 “ 0. Define endomorphisms xα, hα, x´α, α P Π of T via:

(i) hαpbq “ ćαbb.

(ii) x´αpbq “ vpαq b b.

(iii) xαp1q “ 0 and inductively, xα`

vpβq b b˘

“ vpβq b xαpbq ´ δαβcαbb.

for all b P B. To establish L ‰ 0 we show:

1˝. There exists a Lie-homomorphism L Ñ glpT q with rα Ñ rα for all r P tx, hu and α P ˘Π. In particular,hα ‰, xα ‰ 0 and x´α ‰ 0 for all α P Π.

We merely need to verify that the rα fulfill the relations (Rel: a)–(Rel: c) are fulfilled. By (i) the hα,α P Π, act diagonally on T with respect to B and so commute with each other. Thus (Rel: a) holds. Letα, β P Π and b P B. Then


rxα, x´βspbq “ xα`

x´βpbq˘

´ x´β`

xαpbq˘

“ xα`

vpβq b b˘

´ vpβq b xαpbq

“ vpβq b xαpbq ´ δαβcαbb´ vpβq b xαpbq

“ ´δαβcαb

“ δαβhαpbq

and so rxα, x´βs “ δαβhα. Hence (Rel: c) holds.Note that cα,vpβqbb “ cαβ ` cαb and so

rhα, x´βspbq “ hα`

x´βpbq˘

´ x´β`

hαpbq˘

“ hα`

vpβq b b˘

` x´β`

cαbb˘

“ ćα,vpβqbb`

vpβq b b˘

q ` cαb`

vpβq b b˘

“ ´pcαβ ` cαbq`

vpβq b b˘

q ` cαb`

vpβq b b˘

“ ćαβ`

vpβq b b˘

“ ćαβ x´βpbq.

Hence rhα, x´βs “ ćαβ x´β and the second part of (Rel: b) holds.We claim that

hα`

xβpbq˘

“ pcαβ ´ cαbqxβpbq.

If b “ 1, then xβp1q “ 0 and the equation holds. Suppose inductively that the claim holds for b and letγ P Π. Since rhα, x´γs “ ćαγ x´γ we conclude from 2.3.8(a) and the induction assumption that

hα´

x´γ`

xβpbq˘

¯

“`

´ cαγ ` cαβ ´ cαb˘

x´γ`

xβpbq˘

,

and so

hα`

vpγq b xβpbq˘

“ pcαβ ´ cα,γbbq`

vpγq b xβpbq˘

.

If β “ γ, then cαβ ´ cα,γbb “ cαγ ´ cαγ ´ cαb “ ćαb. If β ‰ γ, then δβγ “ 0. Since hαpbq “ ćαbb weconclude in either case that

hα`

δβγcβbb˘

“ pcαβ ´ cα,γbbqδβγcβbb.

Adding the last two displayed equation gives:

hα´

vpγq b xβpbq ` δβγcβb

¯

“`

cαβ ´ cα,γbb˘`

vpγq b xβpbq ` δβγcβb˘

By definition, xβ`

vpγq b b˘

“ vpγq b xβpbq ` δβγcβbb and so

hα´

xβ`

vpγq b b˘

¯

“`

cαβ ´ cα,γbb˘

´

xβ`

vpγq b b˘

¯

So the claim holds. Thus

rhα, xβspbq “ hα`

xβpbq˘

´ xβ`

hαpbq˘

“ pcαβ ´ cαbqxbpbq ´ pćαb xβpbqq “ cαβ xβpbq

and so rhα, xβs “ cαβ xβ. Hence also the first part of (Rel: b) holds and (1˝) is proved.


2˝. Let kα P K, α P Π such that rř

αPΠ kαhα, xβs “ 0 for all β P Π. Then kα “ 0 for all α P Π. In particular,phαqαPΠ is a K-basis for H.

0 “

«

ÿ

αPΠ

kαhα, xβ

ff

“ÿ

αPΠ

kαcαβ xβ “

˜

ÿ

αPΠ

kαcαβ

¸

xβ

By (1˝), xβ ‰ 0 and soř

αPΠ kαcαβ “ 0. Since C is invertible, this implies kα “ 0 for all P Π and (2˝) isproved.

3˝. There exists Q-linear functions ρ : xΦyQ Ñ H˚ and ρ : xΦyQ Ñ H such that ρpdqpρpeqq “ pd, eq for alld P xΦyQ and e P xΦyQ. Moreover,

(a) ρpαqphβq “`

α, β˘

“ cβα for all α, β P Π.

(b) ρpβq “ hβ for all β P Π.

(c) pρpαqqαPΠ is K-basis for H˚. In particular,Ş

αPΠ ker ρpαq “ 0.

(d) ρ and ρ are 1-1.

By (2˝), phβqβPΠ is a K-basis for H. Hence for each α P Π there exists a unique ρpαq P H˚ with

ρpαqphβq “`

α, β˘

“ cβα

for all β P Π. Since Π is a Q basis for xΦyQ, the function Π Ñ H˚, α Ñ ρpαq extends unique to Q-linearfunction ρ : xΦyQ Ñ H˚.

Since pβqβPΠ is a Q basis for xΦy there exists a unique Q-linear function

ρ : xΦyQ Ñ H with ρpβq “ hβ for all β P Π

Then for all α, β P Πρpαqpρpβqq “ ρpαqphβq “

`

α, β˘

.

Note that both pd, eq Ñ ρpdqpρpeqq and pd, eq Ñ pd, eq are Q-bilinear. Since Π and Π are Q-basis forxΦyQ and xΦyQ respectively, we conclude that

ρpeqpρpeqq “ pd, eq

for all d P xΦyQ, e P xΦyQ.Let kα P K, α PP Π with

ř

αPΠ kαρpαq “ 0. Then for all β P Π

0 “ pÿ

αPΠ

kαρpαqqphβq “ÿ

αPΠ

kαρpαqphβq “ÿ

αPΠ

kαcβα “ÿ

αPΠ

cβαka

Since the matrix rcβαsβ,αPΠ is invertible we conclude that kα “ 0 for all α P Π. Thus pρpαqqαPΠ is linearindependent. Since Π is basis for xΦyQ this hows that ρ is 1-1. Also dim H˚ “ dim H “ |Π| and pρpαqqαPΠis a K-basis for H˚. In particular,

Ş

αPΠ ker ρpαq “ 0.Let e P ker ρ. Then for all d P xΦyQ, pd, eq “ ρpdqρpeqq “ ρpdqp0q “ 0. Since p¨, ¨q is non-degenerate

this shows that d “ 0 So also ρ is 1-1 and (3˝) holds.

Let n P Z` and a “ pα1, a2, . . . anq P Πn Y p´Πnq define xa inductively by

xpα1q :“ xα1 and xpα0,aq “ rxα0 , xas


Soxa “ rxα1 , rxα2 , . . . , xαns . . .s.

Also define anda˝ :“ α1 ` α2 ` . . .` αn P ˘xΠyN.

Note that n “ ht a˝. Put X0 “ H. For n P Z` and ε P t`,ú define

Xεn “ xxa | a P pεΠqnqyK

Let µ P ˘NΠ. If µ “ 0 put Xpµq “ H and if µ ‰ 0 put

Xpµq :“ xxa | a P ˘Πht µ, a˝ “ µyK.

4˝. (a) xa P Lρpa˝q.

(b) Xpµq “ Lρpµq.

(c) Let m P Z. Then rXm, X0s ď Xm, rXm, X1s ď Xm`1 and rXm, X´1s ď Xm´1.

(d) L “À

mPZ Xm “À

µP˘xΠyNXpµq “

À

µP˘xΠyNLρpµq.

(e) Let λ be a weight for H on L. Then λ “ ρpµq for some µ P ˘NΠ.

(f) Let α P Π. Then Lρpαq “ Kxα is 1-dimensional, while Lρpkαq “ 0 for all k P Z with k ą 1.

By (Rel: b) , rhβ, xαs “ cβαxα “ ρpαqphβqxα for all α, β P Π and so xα P Lρpαq. Let a P Πn. Thus byinduction on n

xpα,aq “ rxα, xas P rLρpαq, Lρpa˝qs ď Lρpαq`ρpα˝q “ Lρppα,aq˝q.

So (a) holds. In particular, Xpµq ď Lρpµq. Thus rXpµq, Hs ď Xpµq and since Xm “ř

ht µ“m Xpµq,rXm, Hs ď Xm.

If n P Z` the definition of Xn`1 implies Xn`1 “ rXa, Xns. Also rX1, X0s “ rX1, Hs ď X1. From (Rel: b)we have

rX1, X´1s “ Kxrxα, x´βs | α, β P Πy “ Kxhα | α P Πy “ X0.

By induction on n and the Jacobi identity

rX1, Xń´1s “ rX1, rX´1, Xńs ď rX´1, rXń, X1ss ` rXń, rX1, X´1ss

ď rX´1, Xń`1s ` rXń, X0s ď Xn

We proved that rXm, X1s ď Xm`1 for all m P Z. By symmetry also rXm, X´1s ď Xm´1 and so (c) holds.Put X “

ř

mPZ Xm. Then by (c) rX, rαs ď X for all r P x, y, h and α P Π. Since L is generate by the rαwe conclude that X is an ideal in L and in particular, a subalgebra. Since X contains all the rα conclude thatX “ L. Thus

L “ X “ÿ

µP˘NΠ

Xpµq ďà

λP˘ρpNΠq

Lλ.

This easily implies (b), (d) and (e).


Let α P Π and k P Z. Note that k “ ht kα and ak :“ pα, α, . . . , αql jh n

k´times

is the unique element of ˘Πk with

a˝k “ kα. If k ą 1, then xak “ 0 and so Lρpαq “ Xpkαq “ 0. Now xa1 “ xα. If xα “ 0 then alsohα “ rxα, x´αs “ 0, a contradiction to (2˝). Thus Lρpαq “ Kxα is one dimensional. So (f) holds and (4˝) isproved.

5˝. Let α, β, γ P Π with α ‰ β and put vαβ “ xćαβ`1α ˚ xβ. Then rx´γ, vαβs “ 0.

Case 1: γ ‰ α . Then rx´γ, xαs “ 0 and so rxα and rx´γ commute in UpLq. Hence

rx´γ, vαβs “ xćαβ`1α ˚ rx´γ, xβs “ ´δγβ xćαβ`1

α ˚ hβ “ ´δγβcβαxćαβα ˚ xα

If cβα “ 0, this is zero as desired. If cβα ‰ 0, then cαβ ă 0. Thus xćαβα ˚ xα “ xćαβ´1

α ˚ rxα, xαs “ 0 andagain we are done.

Case 2: γ “ α. Note thatKxα`Khα`Kx´α is isomorphic to slpK2qwith Chevalley basis px´α, xα,´hαq.Put k “ ćαβ. Since rx´α, xβs “ 0 and r´hα, xβs “ kxβ. Put vi :“ xi

α

i! ˚ xβ. By 2.5.6(c) we have

x´α ˚ vi “ pk ´ pi´ 1qqvi´1

For k “ i` 1, this gives x´α ˚ vk`1 “ 0. Multiplying with pk` 1q! we get x´α ˚ xk`1α ˚ xβ “ 0. As γ “ α

and k ` 1 “ cαβ ` 1, (5˝) also holds in this case.

6˝. For ε “ ˘ put Xε :“ř8

i“1 Xεi “ xxεα | α P ΠyLie ď L. Let I` be the ideal in X` generate by the vαβ,α, b P Π, α ‰ β. Similarly, define I´. Then Iε is an ideal in L.

Put

I0 :“ 0, I1 :“ xvαβ | α ‰ β P ΠyK and inductively Ik`1 “ xrIk, X1sy.

Since the vαβ are eigenvectors (of weight ρpβ`pćαβ` 1qαq) for H we conclude I1 is invariant under H.Suppose Ik is H invariant. Since also X1 is invariant under H we get from 1.4.12 that also Ik`1 “ xrIK , X1sy

is H invariant. So all Ik, k P N are H-invariant. We claim that rIk, X´1s ď Ik´1. From (5˝) we havervαβ, x´γs “ 0 and so rI1, X´1s “ 0 “ I0. By induction

rIk`1, X´1s Ď xrrIk, X1s, X´1sy ď xrX1, X´1s, Iksy ` xrrX´1, I`k s, X1sy

ď xrH, Iksy ` xrIk´1, X1sy ď Ik

It follows thatř8

i“0 Ik is invariant under X1 and X´1 and so also under L. Thus I` “ř8

i“0 Ik and I` isan ideal in L.

7˝. L “ L{pI` ` I´q

By (6˝), I` ` I´ is the ideal in I generated by the xćαβ`1α ˚ xβ and xćαβ`1

´α ˚ x´β, α, β P Π, α ‰ β. So(7˝) follows from the definition of L and L.

If T is any element or subset of L we denote the image of T in L by T . (4˝)(d)

L “à

mPZXm “ X´ ‘ X0 ‘ X` “ pX´ ‘ X`q ‘ H


Since Iε ď Xε this gives

H X pI` ` I´q “ 0 and H “ HpI` ` I´q{pI` ` I´q – H{H X pI` ` I´q “ H{0 – H

It follows that (2˝), (3˝) and (4˝) remain true for L replaced by L.

8˝. For each α P Π, ad xα and ad x´α are locally nilpotent endomorphism ofL.

Put

Tα :“ tt P L | xnα ˚ t “ 0 for some n P N.u

Let s, t P Tα and pick n,m P N with xnα ˚ s “ 0 “ xm

a ˚ t. By 1.3.4(a) , ad xα is a derivation of L. So wecan apply 4.2.4(a) and conclude that

xn`mα

n` m!˚ rs, ts “

ÿ

i` j“n`m

«

xiα

i!˚ s,

x jα

j!˚ t

ff

If i` j “ n`m, then either i ě n or j ě m. We conclude that xn`mα

n`m! ˚ rs, ts “ 0. So also xn`mα ˚ rs, ts “ 0 and

Tα is a Lie subalgebra of L.Let α ‰ β P Π. Then rxα, x´βs “ 0 and so x´β P Tα. By (Rel: d) x´cαβ`1

α ˚ xβ “ 0 and thus xβ P Tα. Itfollows that hβ “ rxβ, x´βs P Tα. Also

rxα, x´αs “ hα, rxα, hαs “ ´2xα, rxα, xαs “ 0.

Hence

xα ˚ xα “ 0, x2α ˚ hα “ 0, x3

α ˚ x´α “ 0.

So also xα, hα, x´α P Tα. Since L is generated by the xγ, hγ, x´γ as a Lie-algebra this gives Tα “ L and(8˝) holds.

9˝. Let A be an ideal in L, w P WpΦq, and d P xΦyQ. Then dim Aρpdq “ dim Aρpwpdqq.

Suppose first that w “ ωα, α P Π. Then

ρpωαpdqq “ ρpδ´ pd, αqαq “ ρpdq ´ ρpdqphαqρpαq

By (4˝)(d) L “ř

µ˘xΠyNLρpµq and so L is the direct sum of weight spaces of H on L. Note that S :“

xxα, hα, x´αyLie is isomorphic to sl2pKq with Chevalley basis pxα, xa, hαq. By (6˝) both xα and xα actslocally nilpotently on L Also by (4˝)(e), x˘α P L˘ρpαq. This be assumptions of 4.2.6 are fulfilled forpxα, x´α, hα, ρpαq, ρpdq, Aq in place of px, y, h, α, λ,Vq. Thus

dim Aρpdq “ dim Aρpdq´ρpdqphαqρpαq

As seen above ρpωαpdqq “ ρpdq ´ ρpdqphαqρpαq and so (9˝) for w “ ωα.Since WpΦq is generated by ωα, α P Π we conclude that (9˝) holds for all w.

10˝. ρpΦq “ ΦLpHq and dimK Lρpαq “ 1 for all α P Φ.


Let α P Φ. By 3.3.2(i) there exists w P WpΦq and β P Π with α “ wpβq. By (4˝)(f) dim Lρpβ “ 1 anddim Lρpkβq “ 0 for all k P Z with k ą 1. Hence (9˝) implies that dim Lρpαq “ 1 and Lρpkαq “ 0 for all k P Zwith k ą 1. In particular, ρpΦq Ď ΦLpHq.

Conversely, let λ P ΦLpHq. From (4˝)(e) we get λ “ ρpµq for some µ P ˘xΠyN. Let w P WpΦq. By (9˝)

dim Lρpwpµqq “ dim Lρpµqq “ dim Lλ ‰ 0

Thus ρpwpµqq P ΦLpHq and (4˝)(e) implies ρpwpµqq “ ρpµ1q for some µ1 P ˘xΠyN. Hence wpµq P ˘xΠyNfor all w P WpΦq.

3.4.12 now implies that µ “ Qα for some α P Φ. Since µ P ˘xΠyN this gives µ “ kα for some k P Z. As0 ‰ Lλ “ Lρpkαq the previous paragraph implies that k “ ˘1 and so λ “ ρp˘αq P ρpΦq.

11˝. dim L “ |Φ| ` |Π| is finite.

By (4˝)(c) L “À

µP˘xΠyNXpµq and by (4˝)(b) Xpµq “ Lρpµq. If 0 ‰ µ PP ˘xΠyN with Lρµ ‰ 0, then

(10˝) shows that µ P Φ. So

L “à

µP˘xΠyN

Xpµq “ Xp0q ‘à

0‰µP˘xΠyN

Lρpµq “ H ‘à

µPΦ

Lρpµq

By (2˝) phαqαPΠ is K-basis for H and so dimK H “ |Π|. By (10˝) dimK Lρpµq “ 1 and so (11˝) holds.

12˝. SolpLq “ 0.

Let A be an abelian ideal in L. We need to show that A “ 0. Since A is invariant under H, A “

pAX Hq ‘À

αPΦ Aρpαq

Suppose that Aρpαq ‰ 0 for some α P Φ. Since Lρpαq is 1-dimensional we get Lρpαq “ Aρpαq. By 3.3.2(i)there exists w P WpΦq and β P Π with α “ wpβq. By (9˝) dim Aβ “ dim Aα and so Lβ “ Aβ ď A. Sinceωβpβq “ ´β we also get L´β “ A´β ď A. Hence xβ and x´β are in A and sine A is abelian hα “ rxα, x´αs “0, a contradiction.

Hence A X Lρpαq “ 0 for all α P Φ and so A ď H. Moreover, rA, Lρpαqs ď A X Lρpαq “ 0 and soA ď ker ρpαq.

By (3˝)(c)Ş

αPΠ ker ρpαq “ 0. Thus A “ 0 and (12˝) is proved.

13˝. H is a Cartan subalgebra of L.

H is abelian and so nilpotent. Put Y :“ř

µ

À

µPΦ Lρpµq. Then L “ H ‘ Y and AnnYpHq “ H X Y “ 0.Also L{H – Y as an H-module and so AnnL{HpHq “ 0. Recall that NLpHq “ tl P L | rl,Hs ď Hu and soNLpHq{H “ AnnL{HpHq “ 0. Thus NLpHq “ H and H is Cartan subalgebra of L.

14˝. Φ and ΦHpLq are similar rootsytem over Q.

By (10˝), ρpΦq “ ΦHpLq and so also ρ`

xΦyQq “ xΦHpLqyQ. So ρ is an isomorphism of root systems.

The Theorem now follows from (11˝)-(14˝). �

Corollary 4.2.8. Let L and L‚ be a standard, perfect, semisimple Lie algebra over the field K. For ε P t , ‚ulet Hε be a cartan subalgebra for Lε with root system Φε and base Πε . For α P ˘Πε let xεα P Lεα withrxεα, x

ε´αs “ hεα. Suppose that ρ : Φ Ñ Φ‚ is similarity of root systems with ρpΠq “ Π‚. Then there

exists a unique isomorphism of Lie-algebras σ : L Ñ L‚ with σpxαq “ x‚ρpαq

for all α P ˘Π. Moreover,σphβq “ h‚

ρpβqand σpLβq “ L‚

ρpαqfor all β P Φ.


Proof. Let Lε

be the Lie algebra with generators xεα, xε´α, h

ε

α, α P Πε and relations as in 4.2.1. By 4.2.1 Lε

fulfills these relations and so there exists a Lie-homomorphism τε : LεÑ Lε with

τεpxεαq “ xα, τεpxε´αq “ xε´α τεphε

αq “ hεα

for all α P Πε . Since Lε is generated by the xα, x´α, hα, τε is onto. By 4.2.7,

dim Lε“ |Φε | ` |Πε | “ dim L.

and so τε is 1-1. Since ρ is similarity of roots systems, 3.1.21(a) shows that

cαβ “ pβ, αq “´

ρpαq, ˇρpβq¯

“ c‚ρpαqρpφq

.

for all α, β P Π. Thus L and L‚

are defined by the same relations and there exists a Lie-isomorphismτ0 : L Ñ L

‚with

τ0pxαq “ x‚ρpαq, τ0px´αq “ x‚´ρpαq τ0phαq “ h‚

ρpαq

Put σ “ τ‚τ0τ´1. Then

σpxαq “ x‚ρpαq

, σpx´αq “ x‚´ρpαq

σphαq “ h‚α

as desired. Since the xα, x´α generate L, σ is uniquely. �

Bibliography

[Hu] J.E. Humphreys, Introduction to Lie Algebras and Representation Theory, Graduate Text In Mathe-matics 9 Springer-Verlag, New York (1972).

[La] S. Lang, Algebra, Addison-Wesley (1965).

[Sp] T.A. Springer, Linear Algebraic Groups, Second Edition, Birkhauser, Boston (1998).

[St] B. Stellmacher, Lie-Algebren, Lecture Notes (2002).

105

Index

a, 5action, 11algebra, 5

bounded, 73

closed, 93composition series, 26

derivation, 9derived series, 37derived subalgebra, 38discrete, 73

faithful, 12

homogeneous, 26

inner derivation., 10isomomorphism of root systems, 95

jump, 26

Lie algebra, 6

outer derivations, 10

representation, 11

semisimple, 26series, 26simple, 26solvable, 37subideal, 25

universal enveloping algebra, 17

106

lie groups and algebras i lecture notes for mth 91...

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