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RiIC/65/12

INTERNATIONAL ATOMIC ENERGY AGENCY

INTERNATIONAL CENTRE FOR THEORETICAL

PHYSICS

ON NONCOMPACT GROUPS

I. ON CLASSIFICATION OF NONCOMPACTREAL SIMPLE LIE GROUPS AND GROUPSCONTAINING THE LORENTZ GROUP

A. O. BARUT

R. RA.CZKA

/ ' ? "" v

1965PIAZZA OBERDAN

TRIESTE

IC/65/12

INTERNATIONAL ATOMIC ENERGY AGSKCY

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

ON 1JO35FCOMPACT GROUPS

I . CLASSIFICATION OF NONCOMPACT REALSIMPLE LIE GROUPS AND GROUPSCONTAINING THE LORENTZ GROUP

A . 0 . B a r u t

R .

Submitted to the Proceedings of the RoyalSociety

TRIESTE

February 1965

A b s t r a c t

The classification of all simple real Lie groups is

discussed and given explicitly in the form of Tables. In

particular four classes of groups are identified which

contain the Lorentz group. These are the groups which

itavR bilinear forms in real, complex and quart era ionic

spaces invariant, and the unimodular group in n-dimension.

ON 1JONCQMPACT GROUPS

I . CLASSIFICATION OP NONCOItPACT REALSIMPLE LIE GROUPS AND GROUPSCOUTADOTG THE LORENTZ GROUP

In view of the efforts of recent times to combine the

Lorentz group with the internal symmetry groups it is of

interest to classify all groups which generalize the real

Lorentz group. Whereas the classification of compact real

semi-simple groups is well known to physicists in connection

with possible internal symmetries, and has been reviewed in

many places, no comparable study seems to exist in the case

of real simple noncompact Lie groups. In this paper we

discuss first the classification of all real simple Lie

groups and then single out those classes which contain the

Lorentz group. '

I PRELIMINARIES

It is well known that the properties of Lie groups are

intimately connected with the properties of Lie algebras.

To every Lie group there corresponds a Lie algebra, to a

subgroup a subalgebra, to an invariant subgroup an ideal in

the Lie algebra, and so on. It is shown rigorously that all

local properties of Lie groups can be described in terms of

Lie algebras \lj . Thus it is clear that a classification of

all Lie algebras gives us also a classification of all Lie

groups. The first classification of complex Lie algebras was

given by KILLING [_2~\ and later completed by CARTAN P3~\

At this point we must recall the precise definition of a

real Lie algebra and its complex extension.

A real Lie algebra R of dimension n is an n-dimensional

- 1 -

linear vector space on the field of real numbers E with an

operation of "multiplication" (commutator) satisfying the

following conditions:

y, Ut] -o£[x, U ^ + ̂ [y, u

for x, y, u € R and o(, |>> € E

(1)

[x, [ y, u]j + [ y, [ u, x]J + £u, [i, y]j

The complex extension [RJ of the given real Lie algebra

R is the linear vector space of all elements z of the form

z » x + t y ; x, y € R , t =• - 1;

considered over the field C of complex numbers. The Lie

commutator operation fzx, z l in |_RJ is induced by the

corresponding operation in R and is given by

L l*V ^2 J +i |/4» X2J (2)

or

(2')

where C are the structure constants of the given real algebra.

Killing and Cartan have shown that there exist four infinite

sequences of complex Lie algebras denoted by A , B , C and B^IT XI li it

and five exceptional Lie algebras denoted by G^, P , Eg, E_ and

Q . However, the classification of all complex semi-simple Lie

algebras does not lead immediately to a classification of all

serai-simple real Lie algebras. This is due to the fundamental

— 2 —

fact that to a given complex Lie algebra X there correspond,

in general, many real Lie algebras which have X as their complex

extension. For example, the real Lie algebra of the 2 + 1

noncompact Lorentz group with, the conuautation relations

and the real Lie algebra of the compact three-dimensional

rotation group with the commutation relation

; [x2, x3] = x^ ; [xy x ^ - *2 (4)

have the same complex extension, namely the complex Lie algebra A^

Among the real forms of a given complex algebra only one is

compact; all others are noncompact*. The classification problem

of real Lie algebras reduces then to an enumeration of all real

forms of the known classes of complex algebras. This was done

partially by CARTAN ]_A\ who used the theory of symmetrio Riemann

spaces. The first complete classification was given by LARDY [ §j

We shall follow the comparatively simple algebraic approach due

to GAHTMACHER [6^ . We first indicate briefly the procedure and

in the next section discuss in detail the classification.

* Going over to the physically important concept of group

representation we see that a representation of the complex

Lie algebra is also a representation of all its real forms.

But "what is a unitary representation for one real form is not

unitary for the other and vice versa. It is the unitarity

condition for the representations that gives such wide physical

differences between compact and noncompact groups.

We start from the compact real Lie algebra R , i.e., the0

Lie algebra in which beside the commutator j_x, y \ of two

irelements x, y 6 R one can define a scalar productc

"We can always choose in R the "basis vectors in such a way thatc

g , • o T_ . Then we introduce the complex extension X of R .ao v ab c

Prom Cartan's theorem we know that there is a one-to-one

correspondence between the complex extension X of the algebra

and its real compact form R . Prom the definition (2) and (2')

of the Lie commutator jZj» z2 [ ^n ^ w e s e e *na't "kbe structureconstants in X are the same as structure constants in R ,

c

Next we look for all transformations in the complex LievHtO

algebra X which transform the basis 21 in XVa new basis g^ r

i.e., g • => P̂ Zg i in such a way that the structure constants

"5?, defined by

will be real. Gantmacber has proved that a transformation P

which leads from the initial structure constants in X to

inequivalent set of structure constants C. , are given by the

n x n transformation matrix

where S is an involutive automorphism of the compact real form Rc

of the complex algebra X. An involutive automorphism in R isc

a linear? one-to-one transformation of R on itself conserving

the Lie multiplication, i.e.,

S £x, y"] - [ ̂ x, Sy ̂ ; x, y fc R

and fulfilling-

b • 1 ,

c

- 4 -

The metric tensor ft , of the real compact Lie algebra R is

transformed under P =» \ S into the new metric tensor

^, = (Nfi1) (Nfs1). cT « s ..ab ' 'ar bs rs ab

Since an indefinite metric tensor &', cannot be obtained from"Jab

the metric tensor g , - Q , for R by real transformations inab u ab c

R it follows that different involutive automorphisms lead tocdifferent Lie algebras ( \|s^ is in fact a complex transformation}.

This is in fact the intuitive proof of the fundamental theorem

of Cartan; "All different real forms of a given complex semi-

simple Lie algebra X may be obtained as follows: First find

all involutive automorphisms S of the compact form R of X.

Then take the basis composed of the "eigenvectors" of the matrix

S, multiply those vectors of this basis whioh correspond to the

characteristic number -1 by L and leave the remaining vectors of

the basis unchanged. To the basis so obtained there corresponds

a real form of the given complex semi-simple Lie group".

Example: As an illustration of the above method we consider the

compact Lie algebra R>,jEq. (4)J of the three-dimensional rotation

group with

[x; , x

The metric tensor is then

There are four involutive transformations in 2,

(1) - ( N j • S(S) - I " * ) • S(3)

The transformations S/?v and S,,v are not automorphisms because

they do not conserve the Lie commutator j x, y j ; ^(i) transforms

R, into itself and for the involutive automorphism S, •, we obtain

- 5 -

(6) Jand

Hence we obtain from Gartan's theorem

which is just the Lie algebra (3) of the noncompact 2 + 1

Lorentz group.

We must emphasize that in the discussion of involutive

automorphisms ve must consider all inner and outer automorphisms*.

The compact forms of the complex algebras B , C , Gp, F , E 7 and

EQ have only inner automorphisms. Such inner automorphisms lead to

real Lie groups with similar structures. For example, for A all

the real Lie groups are defined as the groups of linear transform-

ations leaving invariant the indefinite hermitian form

no,

so all the corresponding real Lie groups of A differ only in the

signature of the metric tensor of the invariant form in an (n + l)

dimensional complex vector space.

The outer automorphisms lead in general to different real Lie

groups than the inner ones. The existences of outer automorphisms

is most easily investigated in terms of Dynkin diagrams of compact

algebras[8jThe Lynkin diagrams are introduced in the following

manner. Every semi-simple Lie algebra is uniquely described by

the so-called £ -system of real root vectors j_l»Vj* - .

* An inner automorphism is an isomorphic mapping of the group G

into itself by a fixed element x of the group: z' =• ̂ ("2.) » xax~ ,

for all Z G G . It induces an inner automorphism of the corresponding

Lie algebras. In fact all inner automorphisms define the adjoint

group Go All other automorphisms are outer automorphisms.

_ 6 -

For example, the ^ -system for A i s

where 1 e • i is a system of orthogonal vectors of the same

modulus in an n + 1 - dimensional Euclidean space. With every

O -system one can associate a IT-system of vectors A A

which are linearly independent and fulfil the conditions

1° if y^p/^fTT then the scalar product (^t»/? 2)

2° if N J ke 5 then \.k - ^

where £ = - 1 and a^ are non-negative integers. The TT -system

uniby

uniquely defines the'o -system. For A the TT -system is given,

- e n

The Dynkin diagram is associated -with the j$ -vectors hy joiningthe points v, and 3 hy r l ines if cos Lf between £>[ and A . i sequal to (- Y ^ 1 ) (r = 0, 1, 2, 3 ) . The Dynkin diagrams of thenon-exceptional algebras are as follows:

L(.AnJ TO 0 — 0 ~Q

L(BJ 0 „ Un

L(Cn) 0 ft = ®- Q Q

O_ o 0

The lengths of the vectors |3* of a given simple Lie algebra

can assume at most two different values and the ratio is equal to

\P£ . In-the above diagram full circle indicates root of greater

length. jj$]

- 7 -

Since the inner automorphisms transform the algebra into itself

they correspond, to the identity transformation of the Dynkin

diagrams. There is another possibility. For example, if we

subject the 11'-system for A to the transformationn

fi\ ~> & n + 1 -

then the Dynkin diagram i3 invariant under this transformation.

This represents an outer automorphism. Using the properties

of IT -systems we can check immediately the following properties

of outer automorphisms:

1 they exist only for the algebras A (n J> l), I1 and E,,XL li w

2 there are only a finite number of discrete automorphisms,

namely six for 3, and two for each ofn^(n /4/» D_ in r 4,

and B,.

We note that for algebras which are not semi-simple we can have

both the discrete and continuous automorphisms. In the most

interesting case of the Poincare group, for example, we have

the continuous outer automorphism f given by the relation

fx (a, A} - (a, A )

where x 6 3"1 and (a, /\ ) represent the elements of the Poincare

group. The transformation ( •? ) induces in the Lie algebra R

of the Poincare group the transformation P« -5> xP« . We can

immediately see that this transformation is possible . only

because the generators P^ create the commutative invariant

subalgebra of R . So we see that the existence cf outur continuous

automorphism is intimately related to semi-simplicity of the group.

- 8 -

II CLASSIFICATION OF SKAL SIMPLE LIE GROUPS

1. Real Lie Groups Related to A

(a) The existence of an inner involutive automorphism

of the compact real form of k leads to I ZLJ. -2. \ real simple

Lie groups. These groups may be realized as .groups of linear

transformations leaving invariant respectively the Hermitian

forms I 4, 6j

where ^ = O,l r | jkj_lS

For 0 vre obtain the compact group Un+1"

in the remaining

cases we have to do with the noncompact groups. To avoid

am"b i ju i t i e s we "aropose to denote the nonconipact groups whichp

have the form {%) , by HUn+1 where t « 1,n+1

. F 3!_LJLl .

(b) The existence of outer involutive automorphisms leads

to two other groups:

(i) n + 1 — even number. We obtain one group, which can be

realized as the linear quaternion group with n + 3- quaternion2

variables. Elements of representation stjace are vectors inn+1

n "*" ̂ dimensional quaternion space Q 2 and the group

elements are a.. "*" ik. x n. ,+., )- natricesAacting in the Q 4.2 2 A. o

i.e., a = (a , ap, .., an+i), 5* ̂ ( ̂ i k ) where a ̂ and 0 i \c

are quaternions. p|g"T

(ii) The second outer automorphism leads to one group which ca.n

be realized as the group of real linear unimodular transformation

in n + 1 variables. Usually we denote this group by the symbol

SL(n + 1,

2 . Real Lie Groups Related to ^

In this case we have only inner involutive automorphisms.

These automorphisms induce n + 1 different simple real Lie group

which can be realized as the groups of real linear transform-

ations leaving invariant respectively the quadratic forms Jj4, 6\

2 . 2 2 2* " T *2£~ X2e+1 " * ' " X2n+1

- 0 ,1 , . . .n

For | = 0 ire obtain compact rotation groups R̂ - in 2n + 1

dimensions. In the remaining cases we have to do with the non-

compact groups which we denote by KRp where £ = 1, . . . , , n.

3. Real Simple Lie Groups Related to Cn

Eere we have two kinds of inner involutive automorphisms.

The f i r s t one leads to jiL-L-x. 1 real simple groups which may be

realized as the groups of linear transformations leaving

invariant simultaneously the skew symmetric "bilinear form

x l y 2 ~ S2y i """ * * * + X2n-l72n " X2ny2n-i

and the indefinite Hermitian form [A, 6J

t-0.

In the case i = 0 we obtain the compact group us-p,» which

after T.'EYL \jf\ we call the unitary simplectic group. The

intersections with IVu lead to the nonconpact group which we[

denote by ITSp I = 1,....^J . These groups can also be

realized in the space of quaternions leaving the bilinear

quadratic form

invariant, where q are quaternions and q the conjugate

quaternions so that qq is real and positive. The equivalence of

the above quaternionic form to two forms ( 9 ), ( 40 ) can be

verified by using the following representation of the quaternions

and writing q =» 2^ b + ̂ ^ Q w i t h ^ = a^ + (, a^ and

- 10 -

z0 = a, + ia-.0.o3 . It should be remarked that in the above

calculation one must use in the multiplication of quaternions

the rule that z C~o = U~nz • ^e notice furthermore that if in

Eq. (lO1) the quaternions q and Q/ are the same, Q » Q/ ,I 1 <2£ I

then the group defined "by this equation reduces to NTH •

The second type of involutive automorphism leads to only

one group which leaves invariant only the skew symmetric form

(9 ) . This is the noncompact symplectic group Sp

4. Real Lie Groups Related to Dn

Here we have two kinds of inner involutive automorphisms

and n + 2 outer ones, which lead to the following groups2

[4, 6] i(a) [—-—~\ real Lie groups which can "be realized as

the groups of real linear transformations leaving invariant

the quadratic forms2, . 2 2 2

Xl 2t" "x:2g+l"'"~ X2n '

e,-o, i.

For [ = 0 we ootain the compact rotation group R- in 2n

dimensional Euclidean spaoe. The remaining groups are non-

compact and will be denoted by HH2 tm 1 . . . — I2

(b) .One real noncompact simple Lie group which can be

realized as a group of linear transformations in 2n complex

variables, leaving simultaneously invariant the bilinear

form

and the indefinite Herraitian form

- 11 -

(c) The, outer invclutive automorphisms lead to j_-~ v;11" •

real simple noncospact Lie groups which can "be realised as the

group of real transformations in 2n variables leaving invariant

the indefinite quadratic forms

2 2 2xl •*' X2e +1""*" A2n

2 5 , •+• 1We denote these noncompact groupsby ̂ p.,

The classification of the non-exceptional real semi-simple

Lie groups is .exhibited in Table I. In some cases the groups listed

are really not semi-simple, but this is almost a trivial point.

In the case of unitary groups we must take the factor group U n /"LL

in order to obtain a simple group. Similarly, in the case of

quaternion groups one must divide by the one-parameter abelian

A » X ( °~--0- ) in order to get a simple

group.

The Character of the Real Groups

The different real forms of the given complex Lie algebra

can be conveniently characterized by the signature of the metric

tensor 9ab " ^ K ieno"';es ^^e number of negative diagonal

components of q , , and V the number of positive ones then

the quantity

S~ |a-V - 2^-rts r-2Vwill be called the character of the< real group. Prom the Silvester

theorem it follows that real groups having different characters

cannot be isomorphic to each.other. The notion of character is

very useful for the identification of global real groups related

to the same complex algebra. The values of characters of the

non-exceptional groups are given also in Table I.

Among the lowest members of the non-exceptional groups there

are important isomorphisms. These imply a number of isomorphisms

between the corresponding real groups. All isomorphisms between

real simple groups and their character is shown in Table II,

- 12 -

¥e now turn to the discussion of the exceptional

groups.

5. Heal Lie_ Groups Eclated to G?

(a) The compact real simple Lie group can be realized

as the group of real orthogonal transformations in the real

seven-diraensional vector space, leaving invariant the operation

of vector multiplication a x b.

T(a x b)

where the components of a x b are defined by

(n)

a.t

( I = 1, .-., 7; the indices on the right-hand side are to be

reduced to modulus 7).

(b) Besides this compact group there exists a noncompact

one which is defined as the group .df real transformations in

the seven^dimensional real space leaving invariant some indefinite

quadratic form in addition to the operation (^0 of the vector

multiplication U. 6j . Both groups have the dimension 14.

6. Real Lie Groups Related to F.

In this case we have three real Lie groups whose structures

are rather complicated. Cartan has found these structures by-

construction of the so-called normal groups. To describe these

groups we introduce the following denotations [6 1

If T is an infinitesimal transformation of a linear group

defined by the equations

fIV

we shall "write this transformation T in the following form:

where f is an arbitrary differentiate function of the zTie take 26 complex variables

and put

• =P i ii

jjy the dash we shall denote the change of sign of the index,

Then Cartan's normal group is the linear group with the follow

ing infinitesimal transformations.

~ y, •-. >'

0 =r>. v

-f Ju^

= 1, 2, 3, 4, 0 = * : , 3 = - 2 , -,' = 3:£ y>0, it y > 0 ;.:a S.<0, if ; <

- 14 -

If we confine the variables in this group to real values,

we obtain a real simple group.

If we subject the complex var iables X;

to the conditions-B TcP

we obtain another simple real group.

If, finally, we replace the conditions (42.).. by the

conditions

(u)

we obtain a compact group. These three groups have the dimension

52.

7. Real Lie Groups Related to E,

The inner involutive automorphisms lead to the following

three r ea l simple Lie groups.

(a) The compact rea l simple group may be r ea l i zed \&~\

as the group of l i nea r transformations in 27 complex var iables

Xp> U* ' ^ H ~~2*p(p, Q. = 1, 2 , . . . , 6 ) , leaving invariant the

foILowing two fcrms:the cubic form

where r

(p, 1» s, t, u, v) = +1, if the permutation is even,

(p, q., s, t, u, v) =• -1, if the permutation is odd,

and the positive definite Hermitian form

(b) The f i r s t noncompact r ea l simple group may be defined

in the same way as the compact group of the preceding case, with

the only difference that instead of the positive definite

Hermitian form (i3 ) w e must take here the indefinite Hermitian

form LX&l t-— 5 !?

XX(c) The second noncompact real simple group is determined

in the same way as in case (a)? with the only difference that

we must take here instead of Eermitian form (f3) , the form

The outer involutive automorphisms lead to two groups.

(d) The simple real group of linear transformations in

the space of 2 7 complex variables ̂ , (/,, 2 pa a-2p(p, 0. - 1,...,6),

leaving invariant the cubic form [~4t 61

where the variables are subject to the following conditions

(<2.) The second real group can be realized as the group of

linear transformations leaving invariant the cubic form (44)

All these groups have the dimension 78.

8, Real Lie Groups Related to E,-,

In this case we obtain three groups which can be realized as

(a) TheYgrcmP of linear transformations in 56 complex

variables Xpiy ="X^p , ̂ = ~ ^ p h, 1 - 1 , •*•, 8 ) , leaving

invariant the following three forms: the positive definite

Hermitian form

- 16 -

the b i l inea r form

and the 'biquadratic forrn

Z Xp, Xrs Xps X,r

\Here

(P»<ltr»s?t,u,v,w) => +1, if the permutation is even,

(p,q,r,s,t,ufv,w) = -1, if the permutation is odd,

(b) The group of linear transformations in 56 complex variables,

leaving invariant beside the bilinear form (15) and the biquadratic

form (l6) the following indefinite Hermitian form:

")

(c) The group of l inear transformations in 56 complex var iables

, V/ , connected by the r e l a t i o n s

where

which leave the forms (l5) an<3- (l6) invar ian t .

(d) The group of l i nea r transformations in 5^ r ea l parameters

leaving invar iant the b i l i nea r and biquadrat ic forms (l5 ) and ( l 6 ) ,

All these groups have the dimension 133.

9. Real Lie Groups Related to En

In t h i s case we have also three rea l Lie groups. For the r ea l i z a t i on

in terras of the adjoint groups, we re fer CARTM f4 "3. • These groups havethe dimension 248.I l l GROUPS COETTAHraiG THE LORENTZ GROUP

I t i s now easy to se lec t among a l l the r e a l simple groups those

which contain the r ea l Lorentz group. These groups can be c lass i f i ed

as follows 1

- 17 -

(a) Groups defined "by a real indefinite quadratic form

in a real space, N R ^ o r N R 2vv f o r 'K^Z and arbitrary

(b) Groups defined by a real indefinite quadratic form in

a complex space, f\| U (for arbitrary Z if 'K./>M or for (H i 3

« if T\ =

(c) Groups defined by a bilinear indefinite form in a

quaternionic space, ^/ Sp

(d) In addition we have one unimodular group,

and one quaternionic group in quaternianic variables,1

, SL (w^ ' j ̂ /

A c k n o w l e d g e m e n t ,

The authors would like to thank Professor A. Salam and

the IAEA for the hospitality extended to them at.the International

Centre for Theoretical Physics, Trieste.

It is a pleasure to thank C. Fronsdal and J. TTerle for

valuable discussions.

- 1 8 -

REFERENCES

1. L. PONTRIAGIN,l1Topolcgical Groups", Pr inceton (1957). •

2. W. KILLING, Math, Ann. j J 252-288 ( l888) , 33. 1-48 (1889),

34. 57-122 (1889), 3 i 161-189 (1890).

3 . E. CARTAN, These (1894), 2eme ed. Pa r i s ,Vu ibe r t (1933).

4. E. CARTAN, Ann. Ecole Hormale Sup. 3eme s e r i e XXXI, 263-353

(1914), Journal Math. Pures e t app. 8 1-39 (1929).

5 . P. LARDY, Commentarii Matheraatici He lve t i c i 8 189-234 (1935-36)

6. P. GiUraiACHER, Matematicheoku Stornik S_ (47) 101-145 ( l939) 5

^ (47) 218-250 (1939).

7. E. WEYL, "Class ica l Groups'.' Pr inceton ( l939) .

8 . S.B. DTMIU, Amer. Hath, Soo. Transc. I I , Vol. 6

J . JACOBSO1T, Lie Algebras, (1962), A. SA.LAM, Theore t ica lPhysics , Tr ies te (1962).

9. G. RAGAH,"Group Theory and Speotroscopy", Lecture Hotes,

Pr inceton ( l95l ) - r e p r i n t e d CERE 6 l - 8 .

R.S. BEHRESDS, J . DRKITLEBT, C. PROUSDAL, B.W. LEE, Rev. Mod.

Ph. 31 1-27 (1962).

10. C. CHEVALLSY, "Theory of Lie Groups", Pr ince ton (1946).

19 -

TABLE X. Nonexceptional Real groups

Sf>aa

n-t t

complex

n+ i(.even)

TLiri

2.H-H

T'-eal

INVARIANT ^OiQ/y

Quatem'ion yroup In 1^-

Quatern'ion variables

Unimodular Qroup

rnt prov/3

2

7l+2rt

2n +n

OtOi/h

«L ""+1a.

grei/ps in the.* 'class

L z J

1

i

7L-H

-f/u real £'<>*/>

i

m*7f\-zL

s-

* * • < %

o

Ifk

OB

X

t

ft

V

u

V

^

N

IX

•6

cv,

A'BL-B Jl . Isomorphisms of

\°t

~c,•3 t- I -c. t <— j

HZ* ~

2

-2

-6

7)

SL

SL (2,c) - H&1

SL f2.£*J © $U>

0

-2

\

\\

•AC'" <Js

I

•

i

SL [4

- ~ —

•4-

3 U4

"" S U Q © J Ut (p

*

fconfc

rojedive grgyb ofVhe Spe?c£ J

*

-6

3

"4

-3

-5

-15