The Role of Lie Algebras

7/28/2019 The Role of Lie Algebras

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C h a p t e r 8T h e R o l e o f L ie A l g e b r a s

" L o c al" a n d " g lo b a l" a s p e c t s o f L i e g r o u p sI t i s n o w t im e t o b e g i n t h e s y s t e m a t i c s t u d y o f L i e g r o u p s . T h e y w e re i n-t r o d u c e d a t a n e a r l y s ta g e i n C h a p t e r 3 s o t h a t t h e g e n e r a l fe a t u r e s o f t h e i rr e p r e s e n t a t i o n t h e o r y c o u l d b e p r e s e n t e d a t t h e s a m e t i m e a s t h e r e p r e s e n -ta t io n th eo r y o f f in i te g roups . A l th oug h the de f in i t ion o f a l inea r L ie g roupg iven in Sec tion 1 o f Ch ap t e r 3 neces sa r ily invo lved the " loca l" coord ina tes( X l, x 2 , . . . , x n ) w h i c h p a r a m e t r i z e e l e m e n t s n e a r th e i d e n t i ty , t h e e m p h a s i s i nt h e s u b s e q u e n t s e c ti o n s o f t h a t c h a p t e r w a s o n t h e " g lo b a l" p r o p e r t i e s ( t h a ti s, the p rope r t i e s o f the whole g r o u p ) , p a r t i c u l a r l y t h e c o n c e p t o f c o m p a c t n e s sa n d i n t e g r a t i o n o n t h e g r o u p .

In the c lose r s tud y o f L ie g roups bo th the " loca l" and th e "g loba l" a spec t sa r e i m p o r t a n t , b u t i t is f a ir to s a y t h a t m o s t o f t h e i n f o r m a t i o n c o n c e r n i n g t h es t ru c tu re o f a L ie g roup comes f rom the inv es t iga t ion o f i t s " loca l" p rope r t i e s .I t i s the m a in pu rpo se o f th i s chap te r to show how these " loca l" p rop er t i e sa re themse lves de te rm ined by the co r r espon d ing " r eal L ie a lgeb ra" . Th e l inki s p rov ided fo r l inea r L ie g roups by the ma t r ix exponen t ia l func t ion , wh ich i sdesc r ibed in Sec t ion 2, and w h ich leads in tu r n to th e idea o f a "on e-pa ram ete rsubg roup " o f Sec t ion 3. Th e co ncep t o f a r ea l L ie a lgeb ra i s in t rod uce d f i r s tfo r the g roup o f p rop er ro ta t io ns in IR3 , fo r wh ich the a rgumen t ( in Sec t ion4 ) is h e lp e d b y t h e g e o m e t r i c a l n a t u r e o f t h e e l e m e n t s u n d e r c o n s i d e r a ti o n .A t th e s am e t ime the ve ry use fu l and c lose ly r e la ted no t io n o f a "com plex L iealgeb ra" is def ined . For gen era l l inear Lie gro up s a s l ight ly d i f ferent l ine ofa rgumen t i s needed , and th i s i s p rov ided in Sec t ion 5 .

C h a p t e r 9 w i ll b e m a i n l y c o n c e r n e d w i t h i n t r o d u c i n g f o r L i e a l g e b r as m a n yof the ideas p rev ious ly d i s cus sed fo r g roups , an d r e la t ing these to the ana lo -gous p roper t i e s o f l inea r L ie g roups . Aga in the em phas i s w i l l be l a rge ly onthe " loca l" a spec t s o f the g roup s .

C h a p t e r 1 0 is d e v o t e d t o t h e r o t a t i o n g r o u p s i n ] R 3, t o t h e r e l a t e d g r o u p sS O ( 3 ) , O ( 3) a n d S U ( 2 ), a n d t o t h e i r L ie a l g eb r a s. N o t o n l y ar e t h e y i m p o r t a n tf o r t h e i r a p p l i c a ti o n s i n a t o m i c a n d n u c l e a r p h y si c s, b u t t h e i r r e p r e s e n t a t i o n s

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136 GRO UP T HEO RY IN PHYSICSl ie a t th e h ear t of mu ch of the repre senta t ion theo ry tha t fol lows in la terchap te rs .

Wi th Chap te r 11 a t t en t ion beg ins to be concen t ra t ed on the so -ca l l ed"simple" and "semi-simple" Lie a lgebras , which have very im po rta nt physica lapp l i ca t ions , and the s t ruc tu re theo ry o f the semi -simple complex Lie a lgebrasis inves t igated in deta i l . Th e repr ese nta t io n th eory of semi-s imple Lie a lgebrasi s desc r ibed in Chap te r 12 .

I t m ay b e he lp fu l t o an t i c ipa te some o f the d i scuss ion by a l e r t ing the readerto the fac t t ha t , a l though to eve ry Lie g roup the re i s a rea l L ie a lgebra whichis unique (up to i somorphism), in genera l severa l non-isomorphic Lie groupscan correspond to the same real Lie a lgebra . Also , a l though to every rea l Liealgebra the complexi f ica t ion i s unique (up to i somorphism), in genera l severa lnon-isomorphic rea l Lie a lgebras correspond to the same complex Lie a lgebra .

In a l l t he a rgument s tha t fo l low invo lv ing mat r i ces , t he "commuta to r"[A, B] of any two m x m m atr ice s A and B is def ined by

[ A , B ] - A B - B A .

T h e m a t r i x e x p o n e n t i a l f u n c t i o nThe mat r ix exponen t i a l func t ion p rov ides the l i nk be tween a l i nea r L ie g roupand i t s correspo nding rea l Lie a lgebra . I t s def in i t ion and cer ta in of i t s prop-er t ies are s imple genera l iza t ions o f those of the fami l iar expone nt ia l fu nct ionof a rea l o r complex number .D e f i n i t i o n The matrix exponential functionIf a i s an m x m m atr ix , the n e xp a i s the m x m ma tr ix def ined by

ooexp a = 1 + E a j / j ! . (8.1)

j - - 1

T h e o r e m I T h e se ri es for e x p a i n E q u a t i o n (8.1 ) c o nv e rg e s for any m x mm a t r i x a .Proof See, for exam ple , C ha pte r 10, Sect ion 2 , of Cornwel l (1984) .

The fo l lowing example wi l l prove to be very s igni f icant .E x a m p l e I The proper rotation matrices R ( T ) o f S O (3) as matrix exponen-tial functionsCons ide r the 3 x 3 ma t r ix a l de f ined by

0 0 O ]a l = 0 0 1 . ( 8 . 2 )

0 - 1 0


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T H E R O L E O F L IE A L G E B R A S 137T h e n , w i t h a = O a l , a j = ( - -1) (J-1) /20Ja1 for j odd, an d

0 0 O]a J = ( - 1 ) J / 2 0 J 0 1 00 0 1

for j even, so tha t1 0 O ]

e x p ( O a l )= 0 c osO s in O .0 - s i n O c osO

( 8 . 3 )

As noted in C ha pte r 1 , Sect ion 1 , the r ig ht -h and s ide of Eq uat ion (8 .3) speci-f i e s a p roper ro t a t ion th rough an ang le 0 in the r igh t -hand sc rew sense abou tthe ax i s O x (cf . Eq ua t ion (1 .3)) . I t wi ll be dem on st r a te d in Sect ion 4 th atevery mat r ix o f SO(3) can be expressed in ma t r ix exponen t i a l fo rm wi th asui table choice of exponent .

The mul t ip l i ca t ion p roper t i e s o f ma t r ix ex ponen t i a l func t ions a re morecomp l i ca t ed th an those o f exponen t i a l func t ions o f rea l or complex numbers ,as the fo l lowing theorem shows.T h e o r e m I I

(a) I f a and b are any m x m m atr ices tha t c o m m u t e ,(exp a ) (ex p b ) - exp(a + b ) = (exp b ) (ex p a ) . (8.4)

(b) If a and b are m x m mat rice s whose entries a re sufficiently small(exp a ) (exp b ) - - expc ,

where1c = a + b + ~ l [ a, b] + ~ { [ a , [ a, b] ] + [b , [ b, a] ]} + . . . , ( 8 . 5 )

where the inf in i te ser ies in Eq uat i on (8 .5) conta ins co m mu tato rs of in-creasingly h igher order . Thus, in genera l ,

(exp a ) (exp b ) ~- (exp b ) (e xp a ) .Eq ua t ion (8.5) i s known as the "Cam pbe l l -Baker -H ausdorf f fo rmula" .

P r o o f(a) If [a, b] = 0 th en (a + b) j J "!/ ! - so th atY~'~k=0{3 (k (j k) !) } ak b j - k ,

Equat ions (8 .4) fo l low in exact ly the same way as the correspondingresul t s for rea l or complex numbers .


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138 G R O U P T H E O R Y I N P H Y S I C S(b) The p rec is e cond i t ions on the smal lnes s o f the e lemen ts o f a and b r e -

qu i r ed to ensu re the convergence o f the s e r ie s in Eq ua t i on (8 .5 ) , toge the rwi th the comple te expres s ion fo r c , may be found in the o r ig ina l paper s(Cam pbe l l 1897a ,b , Bake r 1905, Hausdor f f 1906). A l l tha t w i ll be donehere is to dem ons t r a te the c o r r ec tnes s o f Eq ua t io n (8.5) to s econd o rde r .F rom Equa t ion (8 .1 ) ( to s econd o rde r )

(exp a) ( exp b ) l a 2= { l + a + ~ + . . . } { l + b + b 2 + . . . }l a 2 ~ ) + . . .= 1 + ( a + b ) + ( ~ + a b + b 2 (8.6)

However , to s econd o rde r ,exp c 1 [a, b] + }= e x p { a + b + ~ . ..

1 [a, b] + }= l + { ( a + b ) + ~ . . .1 1 [a, b] + }2~ { ( a + b ) + ~ . . . + . . .

1 ( ab - ba ) + }= l + { a + b + ~ . ..1+ ~ { a 2 + b a + a b + b 2 + . . . } + . . . , ( 8 . 7 )

f rom which Equa t ion (8 .5 ) fo l lows to s econd o rde r , on equa t ing ther igh t -h and s ides o f Equ a t ion s (8.6) and (8 .7 ) .

T h e o r e m I I I T h e m a t r i x e x p o n e n t ia l f u n ct io n f o rm e d f ro m a n m m m a -t r ix a possesses the fo l lowing proper t ies :

( a ) ( ex pa)* = exp(a* ) .(b) Th e t ra nsp ose of (exp a) i s exp(f i ).( c) ( ex pa) t = ex p(a t ) .(d ) Fo r any m m no n- s ingu la r m a t r i x S

e x p ( S a S - 1 ) = S ( e x p a ) S - 1 .(e) I f A1, A 2, . . . , Am are the e igenvalues of a , th en e ~1, e~ 2 , . . . , e ~m are the

eigenvalues of exp a .( f ) d e t ( e x p a ) = e x p ( t r a ) .(g) exp a i s a lways non-s ing ular and

( ex p a ) - 1 - - e x p ( - a ) .


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T H E R O L E O F L IE A L G E B R A S 139( h) T h e m a p p i n g r = e x p a is a o n e - t o - o n e c o n t i n u o u s m a p p i n g o f a s m a l l

n e i g h b o u r h o o d o f t h e r n x r n z e r o m a t r i x 0 o n t o a s m a l l n e i g h b o u r h o o dof t he m x rn un i t m a t r i x 1 .

Proof S ee, fo r exam pl e , A ppe nd i x E , S ec t i on 1 , o f C ornwel l (1984).

3 O n e - p a r a m e t e r s u b g r o u p sD e f i n i t i o n One-parameter subgroup of a l inear Lie groupA " o n e - p a r a m e t e r s u b g r o u p " o f a l i n e a r L ie g r o u p g is a L ie s u b g r o u p o f gcons i s t i ng o f e l em en t s T ( t ) w h i c h d e p e n d o n a r e a l p a r a m e t e r t t h a t t a k e s a l lv a lu e s f ro m - c ~ t o + c o s u ch t h a t

T ( s ) T ( t ) = T ( s + t) (8.8)for a l l s an d t , - o c < s , t < +o c.

I n p a r t ic u l a r , if g is a g r o u p o f r n x rn m a t r i c e s t h e n a o n e - p a r a m e t e rs u b g r o u p o f g is a L ie s u b g r o u p o f m a t r i c e s A ( t ) s u c h t h a t

A ( s ) A ( t ) = A ( s + t) (s.9)for a l l s and t , - c o < s , t < +c ~ .

C lea r l y T ( s ) T ( t ) = T ( t ) T ( s ) f o r a l l s a n d t , s o e v e r y o n e - p a r a m e t e r s u b -g roup is Abe l i an . M oreover , Equ a t i o n (8 .8 ) w i th s = 0 im p l i e s t h a t T (0 ) = E ,t h e i d e n t i t y of g . O b v i o u s l y a o n e - p a r a m e t e r s u b g r o u p is a L ie g r o u p o f d i-m ens ion 1 , so t ha t i n t he m a t r i x case d A / d t for t = 0 ex i s t s a nd i s no tident ica l ly zero .E x a m p l e I A one-pa ram eter subgroup o f S O ( 3 )T h e 3 z 3 m a t r i c e s A ( t ) d e f in e d b y

A(t) = 1 0 0 ]0 cos t sin t0 - s i n t c o s t

s a t is f y E q u a t i o n ( 8. 9) a n d f o r m a s u b g r o u p o f S O ( 3 ) t h a t is ( b y E q u a t i o n( 3. 7 )) i s o m o r p h i c to t h e L ie g r o u p S O ( 2 ) . T h u s t h e s e m a t r i c e s f o r m a o n e -p a r a m e t e r s u b g r o u p o f S O ( 3 ) . A s s h o w n b y E x a m p l e I o f S e c t i o n 2, A ( t ) =e x p ( t a i ) , w h e r e a i is s p ec if ie d b y E q u a t i o n ( 8 .2 ).

T h e p r o p e r t y e x h i b i t e d i n t h i s e x a m p l e i s c o m p l e t e l y g e n e r a l , a s t h e f o l -l owing t heo rem shows .T h e o r e m I E v e r y o n e - p a r a m e t e r s u b g r o u p o f a l in e a r L i e g r o u p g o f m xm m a t r i c e s i s f o r m e d b y e x p o n e n t i a t i o n o f m x rn m a t r i c e s. I n d e e d , i f t h e


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140 G R O U P T H E O R Y I N P H Y S I C Sm a t r i c e s A ( t ) f o r m a o n e - p a r a m e t e r s u b g r o u p o f ~ , t h e n

A( t ) = exp { ta } , ( 8 .10)w h e r e a = d A / d t e v a l u a t e d a t t - 0.P r o o f F o r b r e v i t y w r i t e A ( t ) = d A / d t , so a = A( 0 ) . L e t

S ( t ) = A ( t ) e x p { - t A ( 0 ) } ,s o t h a t 13 (t) = { / k ( t ) - A ( t ) / k ( 0 ) } e x p { - t / ~ ( 0 ) } . H o w e ve r, f ro m E q u a t i o n(8.9) , for any t ,

.&( t) = l im [ A ( t + s ) - A ( t ) ] / s = l i m A ( t ) [ A ( s ) - A ( 0 ) ] / s ,s--~O s---,Os o t h a t

/ ik(t ) = A ( t ) A (0 ) . (8 .11)T h us ] 3 ( t) = 0 f o r a l l t and consequ en t ly B( t ) - B( 0 ) = 1 , f r om whichE qua t ion ( 8 .10) f o l lows immedia t e ly .

4 L i e a l g e b r a sFor the l i nea r L ie g r oup SO( 3) i t w i l l now be shown tha t i t i s poss ib l e t oi n t r o d u c e t h e c o r r e s p o n d i n g r e a l L i e a l g e b r a i n a v e r y d i r e c t w a y b y a c o m b i -n a t i o n o f a l g e b ra i c a n d g e o m e t r i c a r g u m e n t s . F o r t h e o t h e r l i n ea r L ie g r o u p sthe e s sen t i a l r e su l t s a r e s imi l a r , bu t t he a r gument s a r e r a the r l onge r and l e s sd i r ec t .

I t i s e s sen t i a l t o bea r i n mind tha t , i n t h i s con tex t , t he r e a r e three m u t u a l l yi s o m o r p h i c g r o u p s , n a m e l y

( a) t he g r oups o f a l l p r o pe r r o t a t ion s T in ]R 3 ,( b) t h e g r o u p S O ( 3 ) o f r o t a t i o n m a t r ic e s R ( T ) , a n d(c ) t h e c o r r e s p o n d i n g g r o u p o f l in e a r o p e r a t o r s P ( T ) ( as de fined in E q ua t io n

(1.17)) .Cons ide r f i r s t a n y pr o pe r r o t a t ion T in ] R 3 . Sup pose th i s i s a r o t a t io n

t h r o u g h a n a n g l e ~ 0 a b o u t a c e r t a i n a x i s . T h e n t h e s e t o f all r o t a t i o n s a b o u tt h a t a x i s f o rm a o n e - p a r a m e t e r s u b g r o u p . C o n s e q u e n t l y e v e r y p r o p e r r o t a -t i o n l ie s i n s o m e o n e - p a r a m e t e r s u b g r ou p o f th e g r ou p o f p r o p e r r o t a t i o n s i nIR 3. C o r r e s p o n d i n g l y , e v e r y m a t r i x o f S O ( 3 ) m u s t l ie in s o m e o n e - p a r a m e t e rs u b g r o u p o f s o ( a ) . B y t h e t h e o r e m o f S e c t io n 3, i f A ( t ) a r e t h e e l e m e n t s o fs u c h a o n e - p a r a m e t e r s u b g r o u p o f S O ( 3 ) , t h e r e e x i s ts a n o n -z e r o 3 3 m a -t r i x a s u c h t h a t h ( t ) = e x p ( t a ) . A s h ( t ) is r ea l, T h e o r e m I I I o f S e c t i o n 2i m p l i e s t h a t a a l s o c a n b e t a k e n t o b e r e a l . T h i s t h e o r e m a n d t h e c o n d i t i o nh ( t ) - h ( t ) - 1 a ls o i m p l y t h a t ~ = - a . C o n v e rs e ly , i f a is r e al a n d ~ - - a ,


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T H E R O L E O F L IE A L G E B R A S 14 1t h e n A ( t ) = e x p ( t a ) i s a m e m b e r o f S O ( 3 ) . T h u s every e l emen t o f S O ( 3 ) isob ta ined by exponen t ia t ion f rom some 3 x 3 rea l an t i symmetr i c ma t r i x .

Th e s e t o f a ll 3 x 3 r ea l an t i s y m m et r i c ma t r i c e s fo rms a t h r ee -d i me n s i o n a lr ea l v ec t o r s p a ce ( see A p p en d i x B , Sec t i o n 1 ) . ( I t f o rms a r ea l v ec t o r s p aceb ecau s e , i f a an d b a r e an y t w o s u ch ma t r i c e s , t h en s o t o o is a a + ~b , f o r an yr e a l n u m b e r s a a n d ~ . T h e d i m e n s i o n is t h r e e b e c a u se , f o r a n y s uc h m a t r i xa, a l l = a22 = a33 = 0 an d a21 = --a12, a31 = -a 13 , a nd a23 = -a 32 .)Co n s eq u en t l y a c an b e s p ec i f i ed b y t h r ee r ea l p a r ame t e r s , s u ch a s i t s a1 2 , a1 3and a23 e lem ents . A conven ien t bas i s fo r th i s vec to r space i s fo rm ed by them a t r i c e s[0 0 0 ] [ 0 0 1 ] [ 0 1 0 ]

a l = 0 0 1 , a 2 = 0 0 0 , a 3 = - 1 0 0 . ( 8 . 1 2 )0 - 1 0 1 0 0 0 0 0T h e s e g e n e r a t e o n e - p a r a m e t e r s u b g r o u p s o f m a t r i c es R ( T ) c o r r e sp o n d i n g t or o t a t i o n s a b o u t O x , O y a n d O z r e s p ec t i v e l y ( cf . t h e Ex am p l e s o f Sec t i o n s 2an d 3 ) .

A s t h e c o m m u t a t o r [a , b ] ( = a b - b a ) o f t w o r e a l 3 x 3 a n t i s y m m e t r i cma t r i c e s is a l so a r ea l 3 x 3 an t i s y m m et r i c ma t r i x , [ a, b ] is a me m b er o f t h ev ec t o r s p ace w h en ev e r a an d b a r e mem b er s . Th u s t h e s e t of a ll 3 x 3 r ea lan t i s y m m et r i c ma t r i c e s s a t i sfy t h e co n d i t i o n s i n t h e fo ll ow i n g d e f i n it i o n o f a" r ea l L i e a l g eb ra" .D e f i n i t i o n Real Lie algebra sA "rea l Lie a lgeb ra" s o f d im ens ion n ( _ 1 ) i s a rea l vec to r space o f d im ens ionn eq u i p p ed w i t h a "L i e p ro d u c t " o r " co m m u t a t o r " [a , b] d e f i n ed fo r ev e ry aa n d b o f / : s u ch t h a t

(i) [a, b] C /2 for all a, b E s( ii ) for a l l a , b , c E s an d al l rea l n um be rs c~ an d/ ~

[ca + ~b, c] = a[a, c] + ~[b, c]; (8 .13)(iii) [a, b] = -[ b , a] for all a, b e s an d(iv) for all a , b, c E s

[a, [b, c]] + [b, [c, a]] + [c, [a , b]] = 0. (8 .14)(Th i s is k n o w n a s " J aco b i ' s i d en t i t y " . )

In t h e p a r t i cu l a r c a s e o f a L i e a l g eb ra o f matr i ces t h e c o m m u t a t o r [a , b ]wi l l a lways be defined b y

[ a , b ] = a b - b a , ( 8 . 1 5 )


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142 G R O U P T H E O R Y I N P H Y S I C Sand the n condi t ion s ( i i) , (i ii ) an d ( iv) are au to m at i ca l ly sa t is f ied . S imilar ly , fora Lie a lgebra of l inear opera tors the co m m ut at or [a, b] wi ll a lwa ys be def inedby

[a, b]r = a( br - b(a r (8.16)for any pa i r o f l inea r ope ra to r s a and b and for any e lemen t r o f the ve c to rspace in wh ich the y a c t fo r wh ich the r ig h t -h and s ide of Eq ua t ion (8.16 ) i smean ingfu l .

I t i s not abso lu te ly essent ia l to have an expl ic i t def in i t ion of [a , b] in term sof o the r po s s ib ly s imple r p rod uc t s such as those o f Eq ua t ion (8 .15 ) o r (8 .16 ).Indeed, i t i s an in teres t ing in te l lec tual exerc ise to proceed so le ly on the bas iso f the p rope r t i e s ( i) , ( ii ), ( ii i) and ( iv ) w i thou t m ak ing an y o the r a s sump -t ions abou t the n a tu re o f [a , b]. Th e r esu l t ing deve lopm en t is the theo ry o f"abs t r ac t " L ie a lgeb ras . Th i s does no t l ead to any new s t ruc tu res , for the reex i s t s a theo rem by Ado (1947) wh ich s t a te s tha t every a b s t r a c t L i e a l g e b r ais i somorph ic to a L ie a lgeb ra o f m a t r i ces w i th th e c om m uta to r de f ined asin Eq ua t io n (8 .15 ). Never the les s , much o f the deve lopm en t o f ab s t r ac t L iea lgeb ras i s no more compl ica ted than the co r r espond ing theo ry fo r ma t r i ces ,a n d i n d e e d h a s t h e a d d i t i o n a l a d v a n t a g e t h a t i t a p p l i e s e q u a l l y t o m a t r i c e sand l inea r ope ra to r s a s spec ia l cases . Con seque n t ly the fo rm ula t ion w i l l beg iven in genera l t e rms whenever i t i s conven ien t to do so .

Th i s i s ce r t a in ly the case fo r ce r t a in im m edia te consequences o f the de fi -n i t ion . For exam ple , ( ii ) and ( ii i) im ply tha t[a,/3b + "yc] = ~ [a , b] + "y[a, c]

for all a , b , c E s a n d a ll r ea l n u m b e r s / 3 a n d 7 . F u r t h e r, l et a l , a 2 , . . . , a n b ea bas is of the rea l vector space of /2 . As [ap , aq] E 12 for a l l p , q = 1 , 2 , . . . , n ,

rther e ex is ts a se t of n 3 rea l n um bers Cpq k n o w n a s t h e " s t ruc tu re cons tan t s ofs wi th re spect to the bas is a l , a 2 , . . . , a n " , tha t a r e de f ined by

nl a p , a q ] = rpq ar, p , q = 1, 2 , . . . , n . (8.18)

n

[ a , b ] - E ap~qCpqar. (8.19)p,q,r--1T h u s every co m m uta tor can be eva lua t ed f ro m a knowledge o f t he s truc tu recons tan t s .

In pa r t i cu la r , in the r ea l L ie a l ge b r a / 2 = so (3 ) a s soc ia ted w i th SO(3) ,w i th the bas i s e l emen ts a l , a2 and a3 de f ined by Equa t ions (8 .12 ) ,

[ a l , a 2 ] = - a 3 , [ a 2 , a 3 ] = - a l , [ a 3 , a l l = - a 2 . (8.20)

(Cond i t ions ( iii ) and ( iv ) o f the de f in i tion o f / : imp ly tha t Cpq - -C ~p (forp, q 1 2, n) an d n" " ' ES -- 1 s t s t s t9 , {C pq Crs + CqrCp s -4- CrpCq s } m_ 0 (for p, q, r , t =n1, 2 , . . . , n ) , so t h e s e c o n s t a n t s a r e n o t i n d e p e n d e n t . ) T h e n , i f a = ~ v = l a v a vna n d b - ~ q = l ~ qa q a re a n y t w o e l em e n t s o f s (s o t h a t a l , a 2 , . . . , a ~ a n d

~ , / 3 2 , . . . , ~ = a re al l r ea l ) , by Eq ua t io ns (8 .13 ) , (8 .17) and (8 .18 ) ,


9/17

THE ROLE OF LIE ALG EBR AS 143Consequen t ly the s t ruc tu re cons tan t s w i th r e spec t to a l , a2 and a3 a re g ivenby

1,r - - 1 ,C p q ~ - s - - 0,i f ( p , q , r ) = ( 1 , 2 , 3 ) , ( 2 , 3 , 1 ) , ( 3 , 1 , 2 ) ,if (p, q, r) = (2, 1, 3), (1, 3, 2), (3, 2, 1),for al l other values of (p, q, r ) .

(8.21)

T h e C a m p b e l l - B a k e r - H a u s d o r f f f o r m u l a ( s e e S e c t i o n 2 ) t h e n i n d i c a t e s t h a tthe p rodu c t o f any two e lemen ts o f the group SO(3) ly ing c lose to the ident i tycan be de te rmined ( a t l eas t in p r inc ip le ) f rom the s t ruc tu re cons tan t s . Tha tis , the structure of the group SO(3) close to its identity is specified by thestructure of its corresponding real Lie algebra s (=so (3 ) ) .

The commuta t ion r e la t ions (Equa t ions (8 .20 ) ) t ake a ve ry f ami l i a r fo rmwhen the r ea l L ie a lgeb ra a s soc ia ted w i th th e g roup o f l inea r ope ra to r s P(T)cor respond ing to the g roup o f p rop er ro ta t ion s in ]R 3 i s cons ide red . Le t Tb e t h e r o t a t i o n c o r r e s p o n d in g t o th e m a t r i x e x p ( t a ) o f s o ( a ) , s o t h a t , b yEqua t ion (1 .17) ,

P ( T ) f ( r ) = f ( { e x p ( t a ) } - l r ) = f ( { e x p ( - t a ) } r ) .A s a = l i m t _ , 0 { e x p ( t a ) - 1}/ t , i t i s na tu ra l to define a co r r espond ing l inea rope ra to r P (a ) by the s ame l imi t ing p rocess . Th a t i s, l e t

P ( a ) = l i m { P ( e x p ( t a ) ) - P (1)}/t.t---,OT h u s , f o r a n y fu n c t i o n f ( r ) i n t h e d o m a i n o f P ( a ) ,

P ( a ) f ( r ) = l i m [ f ( { e x p ( - t a ) } r ) - f (r)]/ t .t - - - ,0However , for smal l t ,

f ( { e x p ( - t a ) } r ) f ( { 1 - t a + . . . } r ) = f ( r - t a r + . . . )_~ f ( r ) - t ~ fi g r a d f ( r ) ,

so tha tP ( a ) - - ~ f i g r a d . (8 .2 2)

(Here r and g ra d a re to be in te r p re te d as 3 x 1 co lumn mat r i c es w i th en t r ie sx, y, z and O/Ox, O/Oy, O/Oz r e spec tive ly .) Thus , f rom Eq ua t io ns (8 .12 ),

P ( a l ) = y O / O z - z O / O y , }P ( a 2 ) = z O / O x - x O / O z ,P ( a 3 ) = xO/ Oy- yO/ Ox . ( 8 . 2 3 )

Equa t ion (8 .22 ) imp l ie s tha t[ P ( a ) , P ( b ) ] = P ( f a , b ]) . (8.24)


10/17

14 4 G R O U P T H E O R Y I N P H Y S I C SIn par t icular , by Equat ions (8 .20) , (or d i rec t ly f rom Equat ions (8 .23)) ,

[P ( a l ) ,P ( a 2 ) ] = - P ( a 3 ) , }[ P ( a 2 ) , P ( a 3 ) ] = - P ( a l ) , (8 .2 5)[ P ( a 3 ) , P ( a l ) ] = - P ( a 2 ) .

T h e k e y o b se rv a t i o n i s t h a t t h e q u a n t u m m e c h a n i c a l o rb i t a l a n g u l a r m o -m e n t u m o p e ra t o r s L ~ , L y an d L~ a re j u s t m u l t ip l e s o f P ( a ~ ) , P ( a 2 ) a n dP (a 3 ) . I n f a c t

L x = ( h / i ) P ( a l ) , L y = ( h / i ) P ( a 2 ) , L z = ( h / i ) P ( a 3 ) , (8.26)(see Schi ff (1968)) , so that Equat ions (8 .25) imply the fami l iar angular mo-m e n t u m c o m m u t a t io n r e la ti on s

[Lz , Ly] = ihLz , [Ly, Lz] = ih L x , [nz , Lx] = ihL u . (8.27)There i s there fo re an i n t ima t e connec t i on be tween the qua n tum theory o f an -gu la r m om en tu m and t he g roup o f p roper ro ta t ion s i n IR 3 .

In par t icular , i t wi l l be shown in Chapter 10 , Sect ion 4 , tha t the determi-na t ion o f the bas is func t ions o f the i r reduc ib le rep resen ta t ions o f the g roup o fp roper ro t a t ions in ] R 3 c a n be reduced to the con s t ruc t ion o f the s imul t ane ouse i g e n fu n c t io n s o f t h e o rb i t a l a n g u l a r m o m e n t u m o p e ra t o r s L z and L 2 , wh ereL 2 is def ined by L 2 = L x 2 -f- Ly 2 A- L z 2.T h i s l a t t e r p ro b l e m i s t r e a t e d i n n e a r l y e v e ry b o o k o n e l e m e n t a ry q u a n t u mmec hanics (e .g . Schiff (1968)). These e igenfun ct ions and thei r corres pon dingeigenvalues can be found by solv ing cer ta in d i fferent ia l equat ions (see Chap-ter 10 , Sect ion 4) , but there exis ts a wel l -known, pure ly algebraic m e t h o d o fdetermining the e igenvalues (see Chapter 10 , Sect ion 3) . I t wi l l become ap-paren t th a t i t is mere ly the p ro to type o f a me thod tha t is app l i cab le to therepresenta t ions of a large c lass of Lie a lgebras .

Th e deta i l s of th is deve lopmen t wi ll be g iven in Ch ap ter 10, Sect ion 4 , bu ti t i s conven ien t to no te he re tha t t he a rgument fo r angu la r momentum oper -a tors involves the " ladder opera tors" L+ a nd L_ , def ined b y L = L ~ + i L y ,th a t i s, i t involves the oper a tors P (a~ ) -4- iP(a2) . Th e s igni f icant poin t i s theappe aranc e he re o f the im ag ina ry num ber i , i nd ica t ing th a t i t is u se ful t oexte nd the def in i t ion of a Lie a lgebra to em brace su ch comp lex l inear combi-nat ions . The resul t ing s t ructure i s ca l led a " c o m p l e x Lie a lgebra" .D e f i n i t i o n Complex Lie a lgebra sA "com plex Lie algebra" s of dim ens ion n (>_ 1) is a comp lex vec to r space o fd imen s ion n equ ipped w i th a "Lie p roduc t " o r "com mu ta to r" possess ing thepr op ert i es (i ), ( i i ) , ( ii i) an d (iv) l isted in the defin i t ion of a real Lie algeb ra,except th a t in (ii ) a an d/3 are now any comp lex n u m b e r s .

Equat ions (8 .17) , (8 .18) and (8 .19) apply a lso to complex Lie a lgebras(a l thoug h now the a an d /3 o f (8.17) , t he s t ruc tu re con s tan t s Cpq of (8.18),a n d t h e a l , a 2 , . . . , a n a n d ~ 1, ~ 2 , . . . , / 3 n o f (8 .1 9) m a y b e c o m p l e x n u m b e r s ) .


11/17

T H E R O L E O F L IE A L G E B R A S 145In t he case o f a r ea l L i e a l geb ra o f m a t r i ces o r o f l i nea r ope ra t o r s whose

b a s i s e l e m e n t s a r e l i n e a r ly i n d e p e n d e n t o v er t h e complex f i e ld there i s nod i f f i cu l t y i n "com plex i fy ing" t he r ea l L i e a l geb ra t o p roduce a un ique com plexL i e a l g e b r a o f t h e s a m e d i m e n s i o n . I n t h e se s i tu a t i o n s t h e c o m p l e x v e c t o rs p a c e m a y b e t a k e n t o h a v e t h e s a m e b a s i s e l e m e n t s a s t h e r e a l v e c t o r s p a c e ,b u t , i n t h e c o m p l e x s p a c e , c o m p l e x l i n e a r c o m b i n a t i o n s o f t h e s e b a s is e l e m e n t sa re a l lowed . In f ac t t h i s is t he p roces s a l r eady enco un t e red i n conn ec t i onw i t h t h e a n g u l a r m o m e n t u m l a d d e r o p e r a t o r s . ( F o r t h e m o r e g e n e r a l c a se t h ep roces s i s r a t he r m o re e l abo ra t e . S ee, fo r exam p le , t he de t a i l ed d i s cus s i on o fC h ap t e r 13, S ec t i on 3 , o f C ornw el l (1984) .)

S o m e w h a t p a r a d o x i c a l ly , t h e s t u d y o f c o m p l e x L i e a l g e b r a s i s m o r es t r a i g h t f o r w a r d t h a n t h a t o f r e a l L i e a l ge b r a s. C o n s e q u e n t l y i t i s c o n v e n i e n tt o i nves t i ga t e t he p rop er t i e s o f a l i nea r L i e g roup by f i rs t i n t rod uc in g t h e co r -r e s p o n d i n g r e a l L i e a l g e b r a , a n d t h e n p r o c e e d i n g a l m o s t i m m e d i a t e l y t o t h ea s s o c i a t e d c o m p l e x L i e a l g e b r a .

Th i s s ec t i on wi l l be conc luded wi t h a de f i n i t i on t ha t app l i e s equa l l y t o r ea la n d c o m p l e x L i e a l g e b r a s :D e f i n i t i o n Abelian Lie algebraA L ie alg eb ra s is said to be "A be l ian " i f [a, b] = 0 for al l a , b C s

T h u s i n a n A b e l i a n L i e a l g e b r a a ll t h e s t r u c t u r e c o n s t a n t s a r e z er o . S u c ha L i e a l g e b r a m a y a l t e r n a t i v e l y b e c a l l e d " c o m m u t a t i v e " .

5 T h e r ea l L i e a l g e b r a s t h a t c o r r e s p o n d t og e n e r a l l i n e a r L i e g r o u p s

F o r G = S O ( 3 ) i t w a s e l e m e n t a r y t o d e m o n s t r a t e t h e o c c u r r e n c e o f o n e -p a r a m e t e r s u b g r o u p s , t h e e x i s t e n c e o f t h e c o r r e s p o n d i n g r e a l L i e a l g e b r afo l lowing f rom these . H owever , fo r a general l inear Lie group G i t i s nec-es sa ry t o r eve r se t he o rd e r o f t he a rgu m e n t . F i r s t ( in subsec t i on ( a ) ) it w i llb e s h o w n t h a t f or e v e r y su c h G a c o r re s p o n d i n g r e a l L i e a l g e b r a o f m a t r i c e sex i s t s , and o n ly t he n ( i n subse c t i on (b ) ) w il l t he ex i s t ence and p rope r t i e s o ft h e o n e - p a r a m e t e r s u b g r o u p s o f ~ b e d e d u c ed .

( a ) T h e e x i s t e n c e o f a r e a l L i e a l g e b r a / : fo r e v e r y l in e a rL i e g r o u p G

As a p re l im ina ry , t he e s sen t i a l po i n t s o f t he de f i n i ti on o f a li nea r L i e g roup G o fd im ens io n n g i ven in C h ap t e r 3 , S ec t i on 1 , w i ll be r e -cas t i n t he spe c i a l case i nw h i c h G a c t u a l l y consists of m m m a t r i ces A ( so t h a t T = A a nd r ( T ) = A ) .T h e r e i s a o n e - t o - o n e c o r r e s p o n d e n c e b e t w e e n t h e s e m a t r i c e s l y i n g c l o s e t ot h e i d e n t i t y a n d t h e p o i n t s i n I R n sa t i s fy i ng C ond i t i on (3 .2 ) wh ich de f i net h e m a t r i x f u n c t i o n A ( x l , x 2 , . . . ,Xn) (E ~ ) , f o r a l l ( x l , x 2 , . . . ,Xn) sa t i s fy ingC o n d i t i o n ( 3.2 ). B y a s s u m p t i o n t h e e le m e n t s o f A ( x l , x 2 , . . . , x n ) a r e a n a l y t ic


12/17

146 G R O UP T H E O R Y I N P H Y S IC Sf u n c t i o n s o f X l , X 2 , . . . , X n . T h e n m x m m a t ri c e s h i , a 2 , . . . , a n d e f in e d b y

( a p ) j k - - ( O A j k / O X p ) x l = x 2 . . . . . x ,~ =o ( s . 2 s )( fo r j , k = 1, 2 , . . . , m ; p = 1, 2 , . . . , n ) (c f. E q u a t i o n ( 3 .3 ) ) t h e n f o r m t h e b a s isf o r a n n - d i m e n s i o n a l r e a l v e c t o r s p a c e V .D e f i n i t i o n Analyt ic curve in GL e t x l ( t ) , x 2 ( t ) , . . . , x n ( t ) b e a s e t o f r e a l an a l y t i c f u n c t i o n s o f t d e f i n ed ins o m e i n t e rv a l [0, t o ), w h e re t o > 0 , s u ch t h a t x j (0 ) = 0 fo r j = 1 , 2 , . . . , n , an dt h e p o i n t ( x l ( t ) , x 2 ( t ) , . . . , X n ( t ) ) sa t i s f i es Cond i t ion (3 .2 ) fo r a l l t i n [0 , t0 ) .T h e n t h e c o r r e s p o n d i n g s e t o f m m m a t r i c e s A ( t ) o f G , d e f in e d b y A ( t ) =A ( x l ( t ) , x 2 ( t ) , . . . , x n ( t ) ) , i s s a i d t o fo rm an "an a l y t i c cu rv e" i n G .

A s A ( 0 ) = 1 , e v e r y a n a l y t i c c u r v e s t a r t s f r o m t h e i d e n t i t y o f ~ . T h e r eis n o r e q u i r e m e n t a t t h is s t a g e t h a t a n a n a l y t i c c u r v e m u s t f o r m p a r t o f ao n e - p a r a m e t e r s u b g r o u p o f G .D e f i n i t i o n Tangent vector of an analy t ic curve in GT h e " t a n g e n t v e c t o r" o f a n a n a l y t i c c u r v e A ( t ) i n G is d e f i n ed t o b e t h e m mm a t r i x a , w h e r e a = d A ( t ) / d t ev a lu a te d a t t = 0 . (M ore p rec i se ly , th i s is thet a n g e n t v e c t o r " a t t h e i d e n t i t y " , b u t t h i s e x t r a p h a s e w i ll b e o m i t t e d a s n oo t h e r t a n g e n t v e c t o r s w i l l b e c o n s i d e r e d h e r e . )T h e o r e m I T h e t a n g e n t v e c t o r o f a n y a n a l y ti c c u r v e i n G is a m e m b e r oft h e r e a l v e c t o r s p ac e V h a v i n g t h e m a t r i c e s a l , a 2 , . . . , a n o f E q u a t i o n ( 8. 28 )a s i t s b a s i s. Co n v e r s e l y , ev e ry mem b e r o f V i s t h e t an g e n t v ec t o r o f s o mea n a l y t i c c u r v e i n ~ .

nProo f A s d A ( t ) / d t = E p = l ( O A / O x p ) ( d x p / d t ) , i t f o l l o w s t h en t h a t a =nE p = l &p (0 )ap , w h e re ~p (0 ) = (dxp /d t ) t=o . T h u s a e V .nC o n v e r s e ly , s u p p o s e a = ~ p = l ) ~p ap is a n y m e m b e r o f V . T h e n x j ( t ) =

)~ jt, j = 1, 2 , . . . , n , d e f i n es an an a l y t i c cu rv e t h a t h a s a a s i t s t an g e n t v ec t o r .T h e o r e m I I I f a a n d b a r e t h e t a n g e n t v e c t o rs o f t h e a n a l y t i c c u r ve s A ( t )a n d B ( t ) i n ~ , t h e n [ a, b ] ( - a b - b a ) is t h e t a n g e n t v e c t o r o f t h e a n a l y t i ccu rv e C ( t) i n 6 , w h e re

C ( t ) = A ( v / t ) B ( x / t ) A ( v / t ) - ~ B ( x / t ) - ~ . (8 .2 9)

Proo f T h e o r e m I I o f C h a p t e r 3 , S e c t i o n 1 , i m p l ie s t h a t t h e c u r v e C ( t ) d e f in e db y Eq u a t i o n (8 .2 9 ) i s an analyt ic cu rv e in G . W i t h s - v / t, A ( s ) - 1 + s a +

182 5 ! . . . , a ~8 9 + . . . a n d B ( s ) - 1 + s b + 5 + w h e r e - (d2A/d t2 ) t=o a n d1 a Ib ' - (d2B /dt2) t=o. T h e n , t o s e c o n d o r d e r , A ( s ) - 1 - 1 - s a + s 2 ( a 2 - 5 ) + . . .

a n d B ( s ) - 1 - 1 - s b + s 2 ( b 2 - 8 9 . . . . T h u s , a f te r s o m e a l g e b r a , C ( t ) =1 + s 2 [ a , b ] + . . . , s o t h a t ( d C / d t ) t = o - [ a , b ].


13/17

T H E R O L E O F L IE A L G E B R A S 147T h i s l e a d s i m m e d i a t e l y t o t h e f u n d a m e n t a l t h e o r e m :

T h e o r e m I I I For ev e ry l i near L ie g roup G the re e x i s t s a c orre spond ing rea lL i e a l ge b ra s o f t h e s a m e d i m e n s i o n . M or e p r ec i se ly , i f G has d ime ns io n nt h e n th e m m m a t r i c e s a l , a 2 , . . . , a n d e fi ne d b y E q u a t i o n ( 8 .2 8 ) f o r m abas is for sP r o o f A l l t h a t h a s t o b e s h o w n i s t h a t if a a n d b a r e a n y t w o m e m b e r s o fV , th e re a l v e ct o r s p ac e w i t h b a si s a l , a 2 , . . . , a n o f E q u a t i o n ( 8 .2 8 ), t h e nso i s [ a, b ] . H oweve r , by T he or e m I above , a an d b a r e t an ge n t vec to r s t os o m e a n a l y t i c c u r v es A ( t ) a n d B ( t ) i n 6 . T h e n , b y T h e o r e m I I, [ a, b ] is t h et a n g e n t v e c t o r o f t h e a n a l y t i c c u r v e C ( t ) o f E q u a t i o n ( 8.2 9 ), s o [ a, b ] m u s tb e a m e m b e r o f V .

H a v i n g s h ow n t h a t t h e v e c t o r s p ac e V w i t h b a s is a l , a 2 , . . . , a n is a c t u a l l ya r e al Li e a l g eb r a , h e n c e f o r t h V w i ll b e d e n o t e d b y / : ( as i n t h e s t a t e m e n t o fT h e o r e m I I I a b o v e ) .

I n t h e m a t h e m a t i c a l p h y s i c s l i t e r a t u r e a l , a 2 , . . . , a n a re o ft e n r e fe r r ed t oa s t h e " g e n e r at o r s " o f t h e L ie a l g e b r a / : . W h i l e t h e a b o v e c o n s t r u c t i o n o f / :d e p e n d s e x p l ic i tl y o n t h e p a r a m e t r i z a t i o n o f ~ , i t c a n b e s h o w n t h a t a d if fe r en tp a r a m e t r i z a t i o n m e r e l y p r o d u c e s a L ie a l g e b r a t h a t is i s o m o r p h i c t o / : ( in t h esense o f C ha p te r 9 , Sec t ion 3 ) . T ha t i s, t he r ea l L ie a lgebr a co r r e spo ndin g toa l i nea r L ie g r oup i s e s sen t i a l ly un ique . T h i s w i l l become ve r y c l ea r i n manyc a se s o f i n t e r e st a f te r t h e r o le o f t h e o n e - p a r a m e t e r s u b g r o u p s is d e v e lo p e d i nsubsec t ion ( b ) .

H e n c e f o r t h t h e c o n v e n t i o n w i l l b e a d o p t e d t h a t f o r t h e l i n e a r L i e g r o u p sS U ( N ) , U ( N ) , S O ( N ) a n d s o o n , t h e c o r r e s p o n d i n g r ea l L i e a l g e b r a s a red e n o t e d b y s u ( N ) , u ( g ) , s o ( N ) a n d so o n.E x a m p l e I The real L ie a lgebra s = su(2) of the l i near L ie g roup G = SU( 2) .I t f ol lo w s f r o m E x a m p l e I I I o f C h a p t e r 3 , S e c t io n 1 , a n d f r o m E q u a t i o n ( 8. 28 )t h a t t h e g e n e r a t o r s o f / : = s u ( 2 ) a r e1 L 0 ] 1 [ 0 1 ] 0 ] /8 3 0 /a l = ~ i 0 , a 2 = ~ - 1 0 , a l = ~ 0 - i 's o t h a t , b y d i r ec t c a l c u l a ti o n , t h e b a s ic c o m m u t a t i o n r e la t i o n s a re

[ a l , a 2 ] = - a 3 , [ a 2 , a 3 ] - - a l , [ a 3 , a l l = - a 2 . (8.31)I t w i l l b e o b s e r v e d t h a t a l , a 2 a n d a 3 a r e a l l t r a c e l e s s a n t i - H e r m i t i a n

m atr ices , so s i s th e se t of a ll 2 2 t ra c e le s s a n t i - H e r m i t i a n m a t r ic e s . ( T h i sr e su l t w i l l be de r ived mor e d i r ec t ly in E xample I I be low. )

T h i s e x a m p l e a l s o d e m o n s t r a t e s t h a t t h e m a t r i c e s of a real L ie a lgebr an e e d n o t t h e m s e l v e s b e real, f o r c l ea r ly a l and a3 a r e no t r eal . T h e r ea l i t yc o n d i t i o n o f a re a l L i e a l g e b r a / : r e q u ir e s on ly t h a t t h e e l e m e n t s o f / : b e reall i n e a r c o m b i n a t i o n s o f a l , a 2 , . . . , a n.


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148 G R O U P T H E O R Y I N P H Y S I C ST heor em I I I ha s the f o l lowing conve r se :

T h e o r e m I V E v e r y r ea l L ie a lgebr a i s i somor ph ic to the r ea l L ie a lgebr a o fsome l inea r L ie g r oup .P r o o f See F r euden tha l and de Vr i e s ( 1969) .

( b ) T h e r e l a t i o n s h i p o f t h e r e a l L i e a l g e b r a s t o t h e o n e -p a r a m e t e r s u b g r o u p s o f G

T h e o r e m V E v e r y e l e m e n t a o f t h e re a l L ie a l g e b r a s o f a l i n ea r L i e g r o u pG i s a s s o c i a t e d w i t h a o n e - p a r a m e t e r s u b g r o u p o f G d e f i n e d b y

A ( t ) = e x p ( t a )for - -c~ < t < c~.P r o o f See , f o r example , Appendix E , Sec t ion 2 , o f Cor nwe l l ( 1984) .

Cle ar ly a l l e lem ents of s of the form Aa, wh ere A rang es over a l l r ea l va luesbut a i s f ixed, g ive the s a m e o n e - p a r a m e t e r s u b g r o u p o f G .T h e o r e m V I E v e r y e lemen t o f a l inea r L ie g r oup G in som e s m a l l n e i g h -bo ur h o o d o f i ts id e n t i t y b e l o n g s t o s o m e o n e - p a r a m e t e r s u b g r o u p o f ~ . T h a tis , e v e r y s uc h e l e m e n t o f G c a n b e o b t a i n e d b y e x p o n e n t i a t i n g s o m e e l e m e n tof t he cor r e sp ond ing r ea l L ie a lgebr a .

!P r o o f F o r a n y s e t of n r e a l n u m b e r s ( x~ , x ~ , . . . , x n ) , d e fi n e th e m x m m a t r i x/A ( x ~ , x ~ , . . . , x n ) b y

! IA ( x ~ , x 2, . X n ) ---- ex p{ x~ al + x '9 , 2 a2 + . . . + x n a n } , ( s . 3 2 )a l , a 2 , . . . , a n b e i n g t h e b a si s o f s T h a t A ( x ~ , x ~ , . . . , x~ n) e G i s g u a r a n t e e db y T h e o r e m V . C o n s e q u e n t l y t h e o r ig i n al p a r a m e t e r s x l , x 2 , . . . , x n c a n b e

' w i t h x l = x 2 - . . = x n = 0x p r e s s e d a s a n a l y t ic f u n c t i o n s o f x ~ , x ~ , . . . , X n , - .' = 0. As the Ja co b ia n ( d x j / d X ~ k ) i so r r e s p o n d i n g t o x~ = x~2 = . . . = x n

non- ze r o a t x l - x2 = . . . = xn = 0 , t h i s is a one - to - one ma pp ing be tw eensm a l l ne ighb our h oo ds o f t he two or ig ins . I t fo l lows th a t e v e r y e lem ent o f ~ ins o m e s m a l l n e i g h b o u r h o o d c a n b e e x p re s s e d i n t h e f o r m o f E q u a t i o n ( 8 .3 2 ).

I t is w o r t h n o t i n g t h a t a s t h e s e t o f c o o r d i n a t e s ( x~ , x ~ , . . . , x ~ ) o f E q u a -t ion ( 8 .32) sa ti s fi e s a ll t he cond i t ions o f Cha p te r 3 , Sec t ion 1 , i t p r ov idesa n a l t e r n a t i v e t o t h e o r ig i n a l s et x l , x 2 , . . . , x n , a n d b e c a u s e i t is n e c es s a ri lys imply r e l a t ed to s i t i s c a l l ed a se t o f " canonica l coor d ina t e s" f o r G .

T h e r e r e m a i n s t h e q u e s t i o n o f w h e t h e r t h i s r e s u l t e x t e n d s t o t h e w h o l e oft h e c o n n e c t e d s u b g r o u p o f G . T h e n e x t t h e o r e m s h o w s t h a t t h i s is s o i f ~ is


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T H E R O L E O F L I E A L G E B R A S 149c ompac t , b u t i t is p o s s ib l e t o c o n s tr u c t e x a m p l e s t h a t d e m o n s t r a t e t h a t t h i sn e e d n o t be so i f g is n o n - c o m p a c t . ( Se e, fo r e x a m p l e , E x a m p l e I I I o f C h a p t e r10 , Sect ion 5 , o f Cornwel l (1984)) .T h e o r e m V I I I f g is a c o m p a c t l i nea r L i e g roup , ev e ry e l e m e n t o f t h ec o n n e c t e d s u b g r o u p o f g c a n b e e x p r e ss e d i n t h e f o r m e x p a f o r s o m e e l e m e n ta o f t h e c o r r e s p o n d i n g r e a l L i e a l g e b r a s I n p a r t i c u l a r , if g i s c o n n e c t e d a n dc o m p a c t , e v e r y e l e m e n t o f g h a s t h e f o r m e x p a f o r s o m e a E sP r o o f S e e P r i c e ( 1 9 7 7 ) o r D y n k i n a n d O n i s c i k ( 1 9 5 5 ) .

E v e n f o r c o m p a c t c o n n e c t e d L i e g r o u p s t h i s m a p p i n g n e e d n o t b e o n e -to -one , fo r i t is pos s ib l e t ha t ex p a = exp b w i th a ~ : b . F o r exam ple , fo rg = S O ( 3 ) E q u a t i o n ( 8. 3) sh o w s t h a t e x p ( 0 a l ) = e x p { ( 0 + 2 ~ n ) a i } f or n =+ 1 , + 2 , . . . .

T h e e x p o n e n t i a l m a p p i n g p r o v id e s a d i re c t w a y o f d e t e r m i n i n g t h e r e a l L i ea l g e br a s c o r r e s p o n d i n g t o a n u m b e r o f i m p o r t a n t l in e a r L ie g r o u p s t h a t d o e sn o t r e q u i re a n e x p l i c it p a r a m e t r i z a t i o n . T h e f ol lo w i n g e x a m p l e i l l u s t r a t e s t h em e t h o d .E x a m p l e I I The rea l L ie a lgebra / : = s u ( N ) f o r g = S U ( N ) f o r N >_ 2.L e t e x p ( t a ) b e a n y o n e - p a r a m e t e r s u b g r o u p o f g = S U ( N ) , s o t h a t a is s o m eN x g m a t r i x . A s e x p ( t a ) is r e q u i r e d t o b e u n i t a r y , p a r t s ( c) a n d ( g) o fT h e o r e m I I I o f S e c t i o n 2 i m p l y t h a t

a * = - a . ( 8 .3 3 )M oreover , a s i t i s r eq u i re d t h a t de t ( ex p ( t a ) ) = 1 fo r a l l r ea l t , pa r t ( f ) o f t h a tt he o re m sho ws t ha t ex p ( t r ( t a ) ) = 1 fo r a l l r ea l t . C l ea r l y t h i s is on ly pos s i b l eif

t r a = 0. (8.34)T h u s / : = s u ( N ) i s the se t o f a l l t rac el es s an t i -H erm i t ia n N z N m a t r i c e s .

T h e d i m e n s i o n n o f s ( a n d h e n c e o f g ) c a n b e c a l c u l a t e d a s f o ll ow s . E q u a -t i o n ( 8 . 33 ) i m p l i e s t h a t t h e d i a g o n a l e l e m e n t s o f a m u s t a l l b e p u r e l y i m a g -i n ar y . T a k i n g E q u a t i o n ( 8. 34 ) in t o a c c o u n t , t h e s e t o f d i a g o n a l e l e m e n t s iss p e ci fi e d b y N - 1 r e a l p a r a m e t e r s . ( F o r e x a m p l e , i a i l , i a 2 2 , . . . , i a N - i , N - 1m a y b e t a k e n t o h a v e a r b i t r a r y r e a l v a l u e s , b u t a N N - ~ - - E N s 1 a j j . ) S im i -l a rl y , E qu a t i on (8 .33 ) i m p l i e s t ha t t he " lower" o f f -d iagona l e l em en t s o f a ( i. e.t h e a jk w i t h j > k ) a re c o m p l e t e l y s p e ci fi e d b y t h e c o r r e s p o n d i n g " u p p e r "i ( N 2 N ) u p p e r o ff-= * T h e r e a r ef f -d i a g o n al e l e m e n t s a s ak j - - a j k 9d i a g o n a l e l e m e n t s , a n d a s e a c h h a s a n i n d e p e n d e n t r e a l a n d i m a g i n a r y p a r t ,l (N 2 N ) ( = ( N 2 N )he se t o f a l l o f f -d iago nal e lem ents i s speci f ied by 2 . 3 -r e al p a r a m e t e r s . T h u s a (E s r e q u ir e s ( N 2 - N ) ( N - 1 ) ( = ( N 2 - 1 ))r e a l p a r a m e t e r s , o r , p u t a n o t h e r w a y , t h e r e e x i s t N 2 - 1 l i n e a r ly i n d e p e n d e n tt r a c e le s s a n t i - H e r m i t i a n N x N m a t r i c e s . H e n c e n = N 2 - 1.

In pa r t i cu l a r , n = 3 fo r g = S U(2 ) , and n = 8 fo r g = S U(3 ) .


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150 G R O U P T H E O R Y I N P H Y S I C S

G L ( N , C )G L ( N , l R)S L ( N , )S L ( N , l R)

U ( N )S U ( N )U ( p , q )

S U ( p , q )o ( g , C )

S O ( N , C )O ( N )

S O ( N )

O ( p , q )

S O ( p , q )

S 0 * ( N )S p ( ~ , C )S p ( ~ , ~ )S p ( ~ )

S p( r , s )S U * ( N )

C o n d i t i o n s o n A 6 g

A r e a lde t A - - 1A rea l ,de t A = 1A t = A - i

A t = A - z ,de t A = 1A t g = g A - i

A t g = gA -I,det A ----- 1~ k = A -i{ ~ = A - 1 ,de t A = 1{ ~ k _ _ _ A - 1 ,A real

{ . ~ - - A - 1 ,A rea l ,de t A = 1f ~ k g = g A - i,[ A realA g = g A ,A -1 real,det A = I~ = A - 1 ,A t J A = J~ k J A = J

{ A J A = J ,A r e a lA J A = - J iA t - - Ai k J A -- J ,A t G A = GJ A * = A J ,det A = i

gl (N , 9g l ( N , J R)s l ( N , C )s l (N , IR)

u ( N )s u ( N )u(p , q)

su(p , q)s o ( N , C )s o ( N , 9

s o ( N )

s o ( N )

so(p , q)

so(p , q)

s o* ( N )sp(~, c)sp (~ , ~ )

s p ( N )2s p ( r , s )s u* ( N )

C o n d i t i o n s o n a E E

a rea lt r a = 0a rea l ,t r a = 0a t = - - a

a t = --a,tra=Oat g = --ga

atg = --ga,t r a - - 0

a - - - 8 .a - - - a

~ = --a,a real~----a,a real

~g = --ga,a rea l~ g - - - - g a ,a r ea l{- = --a,a t J -- - Ja~J -- -Ja~J ---Ja,a rea l~ J = - J a ,a t - - a

~ J = - J a ,a t G = - G a

J a * = a J ,t r a = 0

2 N 2N 22N 2 - 2N 2 - 1

N 2N 2 - I

N 2N 2 - 1N 2 - NN 2 - N

89 2 - N )

1 ( N 2 - N )

89 N 2 - N )

1 ( g 2 _ N )

l ( Y 2 - N )N2+N

1 ( N 2 + N )i ( N 2 -4- N )

89 N 2 + N )N 2 - 1

T a b l e 8 .1 : T h e r e a l L i e a l g e b r a s E o f s o m e i m p o r t a n t l i n ea r L ie g r o u p s G .A and a a r e N x N mat r i ces , wh ich a re complex un les s o the rw ise s t a ted ; gis a n N x N d i a g o n a l m a t r i x w i t h p d i a g o n a l e l e m e n t s + 1 a n d q ( = N - p )d ia go na l e lem ents - 1 , p >_ q >_ 1 . In the las t s ix en tr ie s N is even , and J an dG a re the N x N m at r i ce s de fined in Eq ua t ion s (8 .35) and (8 .36 ) .


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T H E R O L E O F L IE A L G E B R A S 151Ta b le 8 .1 l i st s t he de t a i l s o f t he r ea l L i e a l geb ras be long i n g t o a num be r

o f i m p o r t a n t l i n e a r L i e g r o u p s t h a t c a n b e o b t a i n e d t h i s w a y. I n T a b l e 8.1 Ja n d G a r e t h e N N m a t r i c e s d e f in e d b y

0 1N/2 I (8.35)] = - - 1N /2 0 'a n d - - l r 0 0 0

G = 0 Is 0 0 (8.36)0 0 - l r 0 '0 0 0 1 ~

1 N a n d s = 1h e r e 1 _ r _< ~ ~ N - r .T h a t t h e e x p o n e n t i a l m a p p i n g r e m a i n s i n v a l u a b l e e v e n f o r non-compactl i n e a r L i e g r o u p s is d e m o n s t r a t e d b y t h e f ol lo w i ng t h e o r e m .T h e o r e m V I I I Every e l e m e n t o f t h e c o n n e c t e d s u b g r o u p o f any l inear Lieg r o u p G c a n b e e x p r e ss e d a s a finite p r o d u c t o f e x p o n e n t i a l s o f i ts r e a l L iea l g e b r a sProof S ee, fo r exam ple , A pp en d i x E , S ec t i on 2 , o f C ornw el l (1984).

T h e s e r e s u l t s m a y b e summarized b y t h e s t a t e m e n t t h a t t h e m a t r i x ex -p o n e n t i a l f u n c t i o n always p r o v i d e s a m a p p i n g o f s into ~. T h i s i s onto if Gc o n n e c t e d a n d compact, a n d e v e n w h e n ~ is c o n n e c t e d b u t non-compact e v e r ye l em en t o f G i s exp res s ib l e a s a finite p r o d u c t o f e x p o n e n t i a ls o f m e m b e r s o fs

The Role of Lie Algebras

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