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C h a p t e r 3L i e G r o u p sI t is n o w t im e t o f o r m u l a t e a d e f i n it io n o f a L i e g r o u p a n d t o d e s c r i b e s o m eo f t h e m a j o r p r o p e r t i e s o f s u c h g r o u p s . R e a d e r s w h o s e in t e r e s t s l ie o n l y i nt h e a p p l i c a t io n s t o s o l id s t a t e p h y s i c s ( w h e r e o n l y fi n it e g r o u p s a p p e a r ) m a ysa fe ly om i t t h i s chap t e r .

D e f i n i t i o n o f a l i n e a r L i e g r o u pA L ie g r o u p e m b o d i e s t h r e e d i f fe r e n t f o r m s o f m a t h e m a t i c a l s t r u c t u r e . F i r s tl y ,i t s a ti sf ie s t h e g r o u p a x i o m s o f C h a p t e r 1 a n d s o h a s t h e g r o u p s t r u c t u r ed e s c r i b e d in C h a p t e r 2 . S e c o n dl y, t h e e l e m e n t s o f t h e g r o u p a l s o f o r m a" t o p o lo g i c a l s p a c e " , s o t h a t i t m a y b e d e s c r i b e d a s b e in g a s p e c i a l c a se o f a" t o p o lo g i c a l g r o u p " . F i n a ll y , t h e e l e m e n t s a ls o c o n s t i t u t e a n " a n a l y t i c m a n -i f o l d ' . C o n s e q u e n t l y a L ie g r o u p c a n b e d e f i n ed i n s e v e ra l d i f fe r e n t ( b u te q u i v a le n t ) w a y s , d e p e n d i n g o n t h e d e g r e e o f e m p h a s i s t h a t is b e i n g a c c o r d e dt o t h e v a r i o u s a s p e c t s . I n p a r t i c u l a r , i t c a n b e d e f i n e d a s a t o p o l o g i c a l g r o u pw i t h c e r t a i n a d d i t i o n a l a n a l y t i c p r o p e r t i e s ( P o n t r j a g i n 1 9 4 6 , 1 9 8 6 ) o r , a l t e r -n a t i v e l y , a s a n a n a l y t i c m a n i f o l d w i t h a d d i t i o n a l g r o u p p r o p e r t i e s ( C h e v a l l e y1 94 6, A d a m s 1 9 69 , V a r a d a r a j a n 1 9 74 , W a r n e r 1 9 71 ). B o t h o f t h e s e f o r m u -l a t io n s i nv o lv e t h e i n t r o d u c t i o n o f a s e r ie s o f a n c i l l a r y c o n c e p t s o f a r a t h e ra b s t r a c t n a t u r e .

V e r y f o r t u n a t e l y , e v e r y L i e g r o u p t h a t i s i m p o r t a n t i n p h y s i c a l p r o b l e m s i so f a ty p e , k n o w n a s a " li n e ar L i e g r o u p " , f o r w h i c h a re l a t iv e l y s t r a i g h t f o r w a r ddef in i t ion can be g iven . As wil l be s een , t h i s de f i n i t i on is b o t h p rec i s e an ds im p le , in t h a t i t i nvo l ves on l y f am i l i a r concre t e ob j e c t s such as m a t r i c es andc o n t a i n s n o m e n t i o n o f t o p o l o g i c a l s p a c e s o r a n a l y t i c m a n i f o l d s . ( R e a d e r sw h o a r e i n t e r e s t e d i n t h e general d e f i n it io n o f a L i e g r o u p i n t e r m s o f a n a l y t i cm a n i f o l d s m a y , f o r e x a m p l e , f i nd t h is f o r m u l a t i o n i n A p p e n d i x J o f C o r n w e l l(1984) . )

T h e b a s ic f e a t u r e o f a n y L i e g r o u p is t h a t i t h a s a n o n - c o u n t a b l e n u m b e ro f e l e m e n t s l y in g in a r e g i o n " n e a r " i ts i d e n t i t y a n d t h a t t h e s t r u c t u r e o ft h i s re g i o n b o t h v e r y la r g e l y d e t e r m i n e s t h e s t r u c t u r e o f t h e w h o l e g r o u p a n d

35

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36 G R O U P T H E O R Y I N P H Y S I C Sis i ts e l f de t e rm ined by i t s cor re sponding rea l L i e a lgebra . To ensure t ha tt h i s i s so , t he e l ement s i n t h i s r eg ion mus t be pa rame t r i zed i n a pa r t i cu l a rana ly t i c way. Of course , t o say t ha t c e r t a in e lement s a re "nea r" t he i d en t i t ymeans t ha t a no t i on of "d i s t ance" has t o be composed , and i t i s he re t ha t t hecomp l i ca t i ons o f t he g ene ra l t r ea tm en t s t a r t . Howeve r , a ll t he L i e g roup s o fphysica l in te res t a re " l inear" , in the sense tha t they have a t l eas t one fa i thfulf i n it e -d imens iona l r epre sen t a t i on . Th i s r epre sen t a t i on can be used to p rov idethe n ecessa ry p rec i se fo rm ula t ion of d i s t ance and t o ensure t ha t a ll t he o the rtopolog i ca l r equ i rement s a re au toma t i ca l l y obse rved .D e f i n i t i o n L i n e a r L i e g r o u p o f d i m e n s i o n nA gro up ~ is a l inear L ie grou p of dim ension n i f i t sat is f ies the fol lowingcondi t ions (A) , (B) , (C) and (D):

(A) G m u s t p o s s e s s a t l ea s t o n e f a i t h f u l f i n i t e - d i m e n s i o n a l r e p r e s e n t a t i o n r .Suppose t ha t t h i s r epre sen t a t i on has d imens ion m. Then the "d i s t ance"d ( T , T ' ) b e t w e e n t w o e le m e n ts T a n d T ' o f G m a y b e def ined b y

m m

d ( T , T ' ) = + ( E ~ " I r ( T ) j k - r ( T ' ) j k 1 2 }~ / 2 .j = l k = l

(Thi s d i s t ance func t i on d ( T , T ' ) will be ca l led the "m et r ic" . ) T he n(i ) d ( T ' , T ) - d ( T , T ' ) ;

(ii) d ( T , T ) - O ;(iii) d ( T , T ') > O if T ~= T ';(iv) if T , T ' a n d T " a re any t h ree e l ement s o f G ,

d ( T , T " ) < d ( T , T ' ) + d ( T ' , T " ) ,a l l of which a re essent ia l for the inte rpr e ta t io n of d(T , T ~) as a di s tance .(The cho ice o f t h i s me t r i c impl ie s t ha t t he g roup is be ing endow ed w i ththe t opo logy of t he m2-d imens iona l complex E uc l idean space C m2 ( seeEx am ple I I o f A pp end ix B , Sec ti on 2). ) Th e se t o f e l ement s T of G sucht h a t

d ( T , E ) < 6,w here ti is pos i t ive rea l nu m ber , i s the n sa id to " lie in a sphere of rad ius5 cen t red on t he i den t i ty E" , w hich wi ll be denoted b y M~. Such asphe re wi ll be som e t imes re fe r red t o a s a " sma l l ne ighbo urhoo d" of E .

(B ) T here m us t e x i s t a ~ > 0 such t ha t every e l em en t T o f G ly ing i n t hesphere M~ o f rad ius 6 cen t red on t he i den t i t y can be para m et r i z ed byn r ea l p a r a m e t e r s X l , X 2 , . . . , X n ( n o t w o s u c h s e ts o f p a r a m e t e r s c o r re -spond ing t o t he sam e e l em en t T o f G) , t he i den t i t y E be ing par am et r i z edb y x l = x 2 . . . . - x n = O.

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LIEG R O U P S 37Th us eve ry e l ement o f M~ cor re sponds t o one and on ly one po in t i n ann-d imens iona l r ea l Euc l idean space ]Rn , t he i den t i t y E cor re spondingto t h e o r ig in (0, 0 , . . . , 0 ) o f IRn. Moreover , no point in ] a n cor re spondst o m o r e t h a n o n e e l e m e n t T i n M6 .

(C ) The re m us t ex is t a ~7 > 0 suc h that every poin t in IRn fo r wh ichn 2 ~72 (3 .2 )E x j

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38 G R O U P T H E O R Y I N P H Y S I C SI t w ill b e s h ow n i n C h a p t e r 8 t h a t t h e m a t r i c e s a i , a 2 , . . . , a n a c t u a l l y f o rmt h e b a s i s o f a " r e a l L i e a l g e b r a " , a v i t a l o b s e r v a t i o n o n w h i c h m o s t o f t h es u b s e q u e n t t h e o r y is f o u n d e d . H o w e v e r , t h e r e st of t h e p r e s e n t c h a p t e r w i llb e d e v o t e d t o " g r o u p t h e o r e t i c a l " a s p e c t s o f l in e a r L ie g r o u p s .T h e a b o v e d e f i n it io n r e q u i r e s a p a r a m e t r i z a t i o n o n l y o f t h e g r o u p e l e m e n t sb e l o n g i n g t o a s m a l l n e i g h b o u r h o o d o f t h e i d e n t i t y e le m e n t . I n s o m e c a s e s t h i sp a r a m e t r i z a t i o n b y a s in g le s e t of n p a r a m e t e r s x i , x 2 , . . . , x n is v a l id o v e r al a r g e p a r t o f t h e g r o u p o r e v e n o v e r t h e w h o l e g r o u p , b u t t h i s i s n o t e s s e n t ia l .I n S e c t io n 2 it w il l b e s h o w n t h a t t h e w h o l e o f t h e " c o n n e c t e d " s u b g r o u p o f al in e a r L ie g r o u p o f d i m e n s i o n n c a n b e g i v e n a p a r a m e t r i z a t i o n i n t e r m s o f as in g le s e t o f n r e a l n u m b e r s w h i c h w i l l b e d e n o t e d b y y i , y 2 , . . . , y n . H o w e v e r ,t h i s l a t t e r p a r a m e t r i z a t i o n is n o t r e q u i r e d t o s a ti s fy a ll t h e c o n d i t io n s o f t h ea b o v e d e f i n i t i o n , a n d s o n e e d b e a r l i t t l e r e l a t i o n t o t h e p a r a m e t r i z a t i o n b yX i ~ X 2 ~ 9 9 9 ~ Xn .

T h e f o l l o w i n g e x a m p l e s h a v e b e e n c h o s e n b e c a u s e t h e y i l l u s t r a t e a l l t h ee s s e n ti a l p o i n t s o f t h e d e f in i ti o n w i t h o u t i n v ol v in g a n y h e a v y a l g e b r a .E x a m p l e I T h e m u l t ip l i c a t i v e g r o u p o f r ea l n u m b e r sA s in E x a m p l e I o f C h a p t e r 1, S e c t i o n 1 , l e t g b e t h e g r o u p o f r e a l n u m b e r s t( t ~- 0 ) w i t h o r d i n a r y m u l t i p l i c a t io n a s t h e g r o u p m u l t i p l ic a t i o n o p e r a t i o n , t h ei d e n t i t y E b e i n g t h e n u m b e r 1. g h a s t h e o b v io u s o n e - d i m e n s i o n a l f a i t h fu lr e p r e s e n t a t i o n F ( t ) = [ t ], s o c o n d i t i o n ( A ) i s s a ti sf ie d a n d t h e m e t r i c d ofE q u a t i o n ( 3 .1 ) is g i v e n b y d ( t , t ' ) = I t - t ' l . I n p a r t i c u l a r , d ( t , 1 ) = I t - 1 I.iL e t 5 = ~ s o t h a t 89< t < 2 f o r a l l t i n M ~ . A c o n v e n i e n t p a r a m e t r i z a t i o n f ort E M ~ i s t h en

t - ex p x i . (3 .4 )A s r e q u i r e d i n ( B ) , t h e i d e n t i t y 1 c o r r e s p o n d s t o x i = 0 . C o n d i t i o n ( C ) iso b e y ed w i th ~ = l o g 3 , a s x 2 < ( l o g 3 )2 im p l i e s 2 < ex p x i < 3 . B y E q u a t i o n( 3.4 ) F ( x i ) = e x p x i , w h i c h is c e r t a i n l y a n a l y t i c , so t h a t c o n d i t i o n ( D ) iss a t is f ie d . T h u s g i s a l i n e a r L ie g r o u p o f d i m e n s i o n 1 . I t s h o u l d b e n o t e d t h a tE q u a t i o n ( 3. 3) i m p l ie s t h a t a i = [ 1 ], t h e r e b y c o n f i rm i n g t h e f i rs t th e o r e ma b o v e .

I t is s i g ni f ic a n t t h a t t h e p a r a m e t r i z a t i o n i n E q u a t i o n ( 3.4 ) e x t e n d s t o a llt > 0 ( w i t h - o c < x i < + o c ) a n d t h a t t h i s s e t f o r m s a s u b g r o u p o f g .M o r e o v e r , e v e r y g r o u p e l e m e n t t s u c h t h a t t < 0 c a n b e w r i t t e n i n t h e f o r mt = ( - 1 ) e x p x i f or s o m e x i .

E x a m p l e I I T h e g r o u p s 0 ( 2 ) a n d S O ( 2 )0 ( 2 ) is t h e g r o u p o f a l l r e a l o r t h o g o n a l 2 2 m a t r i c e s A , S O ( 2 ) b e i n g t h es u b g r o u p f or w h i c h d e t A = + 1 .

I f A E 0 ( 2 ) , F ( A ) = A p r o v i d e s a f a i th f u l fi n i te - d i m e n s i o n a l r e p r e s e n t a -t io n . T h e o r t h o g o n a l i t y c o n di ti o n s A A = A A = 1 r e q u ir e t h a t

( A l l ) 2 d- ( A i 2 ) 2 - - ( A i i ) 2 + ( A 2 i ) 2 = ( A 2 i ) 2 + ( A 2 2 ) 2- ( A i 2 ) 2 + ( A 2 2 ) 2 - i (3 .5)

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L I E G R O U P S 39a n d A l i A 2 1 + A 2 2 A 1 2 = A l i A 1 2 + A 2 2 A 2 1 = 0 . ( 3 . 6 )E q u a t i o n s ( 3 .5 ) i m p l y t h a t ( A l l ) 2 = ( A 22 ) 2 a n d ( A 12 ) 2 = ( A 21 ) 2 , s o t h a tt h e r e a r e o n l y t w o s e t s o f s o l u t i o n s o f E q u a t i o n s ( 3 .6 ) , n a m e l y :

( i) A l l = A 2 2 a n d A 12 = - A 2 1 . E q u a t i o n s ( 3 . 5 ) i m p l y t h a t d e t A = + 1 , i.e .A e S O ( 2 ) . M o r e o v e r , f r o m E q u a t i o n s ( 3 .5 ) , d ( A , 1 ) = 2 ( 1 - A l l ) 1 /2 .

( ii ) A l l - - A 2 2 a n d A 12 = A 2 1 . I n t h i s c a s e d e t A = - 1 a n d d ( A , 1 ) = 2 .W i t h t h e c h o i ce fi = v /2 , c o n d i t i o n ( B ) r e q u i r e s t h e p a r a m e t r i z a t i o n o f p a r to f s e t ( i) b u t i t i s n o t n e c e s s a r y t o i n c l u d e s e t ( ii ), a s i t is c o m p l e t e l y o u t s i d eM ~ . A c o n v e n i e n t p a r a m e t r i z a t i o n is

[ c o s x l s i n x l I ( 3.7 )t = r( A) = - s i n x l c o s x l "C l e a r l y x l = 0 c o r r e s p o n d s t o t h e g r o u p i d e n t i t y 1 a n d t h e d i m e n s i o n n is 1.

E v e r y p o i n t o f I R 1 s u c h t h a t x ~ < ( 7~ /3 ) 2 g i v e s a m a t r i x A i n M ~ , s o c o n -d i t i o n ( C ) is s a t is f i ed . I n f a c t t h e p a r a m e t r i z a t i o n o f E q u a t i o n ( 3. 7) e x t e n d st o t h e w h o l e o f t h e s e t ( i) w i t h - ~

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40 G R O U P T H E O R Y I N P H Y S I C Sfor th en x l - x2 - x3 - - - - 0 c o r r e s p o n d s t o t h e i d e n t i t y 1 , a n d

d( u , 1) = 211 - {1 - (1 /4 ) ( x 2 x 2 + x2)}1/2] 1/2,so tha t d (u , 1 ) < ~ i f an d on ly i f x 2 + x 2 + x 2 < {2~ 2 1 4 1 /2Z ~ i } . T h u s , w i t h

1~4 con d i t i ons (S ) an d (C ) a re sa t i s f ied and M6 an d2 v /2 a n d r / < 2 ~ 2 - ~ ,R n co inc ide . C o nd i t i on (D) i s c l ea r l y t rue , so S U(2 ) i s a l inea r L i e g ro up o fd i m e n s i o n 3 . I n c i d e n t a l l y , E q u a t i o n ( 3 . 3 ) g i v e s1 [ 0 i f 0 1 ] 1 [ / 0 ] /3 1 0 /a l = ~ i 0 , a 2 = ~ - 1 0 , a 3 = ~ 0 - i ' "s o t h a t t h e f ir st t h e o r e m a b o v e i s y e t a g a i n c o n f i r m e d .

A l t h o u g h t h i s p a r a m e t r i z a t i o n is t h e m o s t c o n v e n i e n t f o r e s t a b li s h in g t h a tS U(2 ) i s a l i nea r L i e g roup , i t i s no t t he m os t u se fu l fo r som e p rac t i ca l ca l -cu l a t i ons . Inde ed on ly t he m a t r i ce s u w i th c~1 >_ 0 can be p a ra m et r i ze d t h i sw a y , w h e r e a s i t w i l l b e s h o w n i n E x a m p l e I I I o f S e c t i o n 2 t h a t t h e r e e x i s tp a r a m e t r i z a t i o n s o f t h e w h o l e o f S U ( 2 ).

T h e r e i s n o d i f f i c u l t y i n p r i n c i p l e i n g e n e r a l i z i n g t h e a r g u m e n t s u s e d i nE x a m p l e s I I a n d I I I t o s h o w t h a t f o r a ll g >_ 2 , o ( g ) , S O ( N ) , V ( N ) a n d1 ( g 1) 8 9 1) g 2 an dU ( N ) a r e l in e a r L i e g r o u p s o f d i m e n s i o n s 5 N - ,N 2 - 1 r e s p e c ti v e ly , b u t t h e d e t a i l e d a l g e b r a i s r a t h e r m o r e l e n g th y . ( U ( 1 ) i sa s p e c ia l c a se t h a t is v e r y e a s y t o t r e a t a l o n g t h e l in e s o f E x a m p l e I , b e c a u s eu = [ ex p ix l ] , -T r < X I __< 7 1 " , i s a p a r a m e t r i z a t i o n . )

F i n a l l y , a L i e s u b g r o u p c a n b e d e f i n e d i n t h e o b v i o u s w a y .D e f i n i t i o n L ie subgroup o f a l i near L i e g roupA sub gro up G ' o f a l i nea r L i e g roup G t h a t i s i t s e l f a l i nea r L i e g roup i s ca l l eda "L i e subg rou p" o f G .

2 T h e c o n n e c t e d c o m p o n e n t s o f a l i n e a r L i eg r o u p

D e f i n i t i o n C o n n e c t e d c o m p o n e n t o f a l i n e a r L i e g r o u pA m a x i m a l s e t of e l e m e n t s T o f G t h a t c a n b e o b t a i n e d f r o m e a c h o t h e r b yc o n t i n u o u s l y v a r y i n g o n e o r m o r e o f t h e m a t r i x e l e m e n t s F ( T ) j k o f t h e f a i t h fu lf i n i te - d i m e n s i o n a l r e p r e s e n t a t i o n r is s a i d t o f o r m a " c o n n e c t e d c o m p o n e n t "of G.

( I t c a n b e s h o w n t h a t t h e c o n c e p t o f c o n n e c t e d n e s s a s d e f i n e d fo r a g e n e r a lt o p o l o g i c a l s p a c e ( S i m m o n s 1 9 63 ) is e q u i v a l e n t, f o r t h e t y p e o f s p a c e b e i n gc o n s i d e r e d h e r e , t o t h a t i m p l i e d b y th e a b o v e d e f i n i ti o n . )E x a m p l e I T he mu l t i p l ica t i ve g roup o f r ea l num ber sT h i s g r o u p w a s c o n s i d e r e d in E x a m p l e I of S e c t i o n 1. T h e s e t t > 0 f o r m s

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LIE GROUPS 41o n e c o n n e c t e d c o m p o n e n t ( w h i c h a c t u a l l y c o n s t i t u t e s a s u b g r o u p ) a n d t h e se tt < 0 f o rm s a n o t h e r c o n n e c t e d c o m p o n e n t . A s t = 0 is e x c l u d e d f r o m t h eg r o u p , o n e s e t c a n n o t b e o b t a i n e d c o n t i n u o u s l y f r o m t h e o t h e r .E x a m p l e I I The groups 0 ( 2 ) and S O ( 2 )I n th e g r o u p 0 ( 2 ) t h a t w a s e x a m i n e d i n E x a m p l e I I o f S e c ti o n 1 , e v e r y m a t r i xA o f S O ( 2 ) c a n b e p a r a m e t r i z e d b y E q u a t i o n ( 3 .7 ) w i t h - T r

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42 G R O U P T H E O R Y I N P H Y S I C Sw h e r e

T h u s0 ~ Yl _~ 7r/2 , 0 ~ Y2 _< 2z r, 0 < Y3 _< 2zr. (3 .11)

cos y l exp ( i y2 )u = r ( u ) = - s in y l e x p ( - i y 3 ) s i n y l e x p ( i y 3 ) I ( 3 1 2 )cos y i ex p ( - i y2 ) '

who se m a t r i x e l em en t s axe obv i ous ly con t i n uou s fun c t i ons o f Y l, Y2 an d Y3.T h i s i s t h e r e f o r e a p a r a m e t r i z a t i o n o f t h e w h o l e o f S U ( 2 ) . ( T h i s p a r a m e t r i z a -t i on f a il s t o s a t i s fy t h e cond i t i ons i nvo l ved in t he de f i n i t i on o f a l i nea r L i eg r o u p b e c a u s e i t d o e s n o t p r o v i d e a one-to-one m a p p i n g o f t h e a p p r o p r i a t ereg ions , fo r t he i de n t i t y co r re spon ds t o t he w ho l e s e t o f po i n t s y l - 0 , Y2 - - 0,0 _~ Y3 __ 27r. Consequently o r / O y 3 = 0 at yl = y2 = y3 = 0.)

S i m il a r a r g u m e n t s s h o w t h a t S O ( N ) a n d S U ( N ) a r e c o n n e c t e d li n e ar L ieg ro ups fo r a l l N > 2 , a s is U (N ) fo r a ll N >_ 1 . T he r e l a t i on sh ip b e t w een ac o n n e c t e d l in e a r L i e g r o u p a n d i ts 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 w i ll b e s t u d -i e d i n s o m e d e t a i l i n C h a p t e r 8 , w h e r e i t w i l l b e s h o w n t h a t t h e L i e a l g e b r av e r y l a r g e ly d e t e r m i n e s t h e s t r u c t u r e o f t h e g r o u p . I n d e e d , i t is f o r t h i s p u r -p o s e t h a t t h e p a r a m e t r i z a t i o n in t e r m s o f x l , x 2 , . . . , x n i s r e q u ir e d . H o w e v e r ,t h e r e s t o f t h i s c h a p t e r is d e v o t e d t o c e r t a i n " g lo b a l" p r o p e r t i e s o f l i n e a r L i eg r o u p s , a n d f or th e s e i t is t h e p a r a m e t r i z a t i o n i n te r m s o f Y l, y 2 , . . . , y n t h a ti s re levant .

3 T h e c o n c e p t o f c o m p a c t n e s s f o r l i n e a r L i eg r o u p s

A l t h o u g h t h e c o n c e p t o f a " c o m p a c t " s e t in a g e n e r a l t o p o l o g i c a l s p a c e h a sa cu r ious ly e lu s ive qua l i t y , t he fo l l owi ng t heo rem , o f t en r e fe r r ed t o a s t he" H e in e - B or e l T h e o r e m " , p r o v id e s a v e r y s t r a i g h t f o r w a r d c h a r a c t e r i z a t i o n o fs u c h s e t s i n f i n i te - d i m e n s i o n a l r e a l a n d c o m p l e x E u c l i d e a n s p a c es . A s t h i s w i lls u f f i c e t o d i s t i n g u i s h a c o m p a c t l i n e a r L i e g r o u p f r o m a n o n - c o m p a c t l i n e a rL i e g r o u p , n o a t t e m p t w i ll b e m a d e t o g iv e a d e t a i le d a c c o u n t o f c o m p a c t n e s s ,n o r e v e n a d e f i n it i o n o f t h e n o t io n . ( A l u c id a c c o u n t o f t h i s a n d o t h e r g e n e r a lt o p o l o g i c a l id e a s m a y b e f o u n d i n t h e b o o k o f S i m m o n s ( 1 9 63 ) .)

T h e o r e m ! A su b s e t o f p o i n ts o f a r e al o r c o m p l e x fi n it e -d i m e n s io n a l E u -c l i dean spac e i s "co m pac t " i f and on ly if i t is closed a n d bounded.A s m e n t i o n e d i n S e c t i o n 1 , b y i n t r o d u c i n g t h e f a i t h f u l m - d i m e n s i o n a l r e p -

r e s e n t a t i o n F , t h e L i e g r o u p h a s b e e n e n d o w e d w i t h t h e t o p o l o g y o f C m 2.H o w e v e r , i t is o f t e n h e l p f u l t o in v o k e t h e c o n t i n u o u s p a r a m e t r i z a t i o n o f t h ec o n n e c te d s u b g r o u p b y y l , y 2 , . . . , Yn i n t r o d u c e d i n S e c t i o n 2 . A s t h e c o n t i n u -o u s i m a g e o f a c o m p a c t s e t is a l w a y s a n o t h e r c o m p a c t s e t ( S i m m o n s 1 9 6 3 ) , i t

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L I E G R O U P S 43f o ll ow s t h a t i f t h e l in e a r L i e g r o u p h a s o n l y a f in it e n u m b e r o f c o n n e c t e d c o m -p o n e n t s a n d t h e p a r a m e t e r s y l , Y 2 , . . . , Yn r a n g e o v e r a c l o s e d a n d b o u n d e ds e t i n I R ~ , t h e n t h e g r o u p is c o m p a c t .

A " b o u n d e d " s e t o f a re a l o r c o m p l e x E u c l i d e a n s p a c e i s m e r e l y a se tt h a t c a n b e c o n t a i n e d i n a f i n i t e " s p h e r e " o f t h e s p ac e . T h e t e r m " c lo s ed "i m p l ie s s o m e t h i n g m o r e in v o l v e d , s o p e r h a p s a f ew w o r d s o f e x p l a n a t i o n m a yb e n e e d e d . A l t h o u g h t h e s p e c i f i c a t io n o f a g e n e ra l c lo s e d s e t c a n b e f a ir l yd i f fi cu lt , t h e on l y sub se t s o f IRn t h a t a r e r e l e v a n t h e r e a r e c o n n e c t e d , a n d f o rt h e s e t h e c h a r a c t e r i z a t i o n is s tr a i g h t f o r w a r d . I n d e e d , i n ]R 1 e v e r y c o n n e c t e dclosed se t i s o f the form a l

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44 G R O UP T H E O R Y I N P H Y SICSfor all N > 2.E x a m p l e I I I The groups U (N ) and S U ( N )As a l l the in terva ls in C on di t io ns (3 .11) are c losed and f in ite , SU (2) is compact.Th e s am e is t rue o f SU (N ) fo r a l l N > 2 , an d o f U (N ) fo r a l l N > 1 .4 I n v a r i a n t i n t e g r a t i o nI f t o e a c h e le m e n t T o f a g r o u p g a c o m p l e x n u m b e r f ( T ) i s a s s igned , thenf ( T ) i s s a id to be a "com plex -va lued func t io n de f ined on g" . One exam plet h a t h a s b e e n m e t a l r e a d y i s t h e s e t o f m a t r i x e l e m e n t s F ( T ) j k ( fo r j , k f ix e d)o f a m a t r i x r e p r e s e n t a t i o n F o f g .For a finite g r o u p s u m s o f t h e form ETEg f (T) a r e f r equen t ly encoun te red ,p a r t i c u l a rl y i n r e p r e s e n t a ti o n t h e or y . B e c a u s e th e R e a r r a n g e m e n t T h e o r e ms h o w s t h a t t h e s e t {T 'T; T E g} has exac t ly the s am e mem bers a s G , i t f o llowst h a t f o r a n y T ' E

E f ( T ' T ) = E f ( T ) ,T E g T E g

and th e sum i s s a id to be " l e f t - inva r ian t " . S imi la r lyE f ( T T ' ) = E f ( T ) ,T 6 g T E g

so such sum s a re a l so " r igh t - inv a r ian t " . M o reover , w i th f ( T ) = 1 for al lT E G, the su m is f in i te in the sense th a t ~-'~Teg 1 - - g , the ord er of G.

In genera l i z ing to a connec ted l inea r L ie g roup , i t i s na tu ra l to make theh y p o t h e s i s t h a t t h e s u m c a n b e r e p l a c e d b y a n i n t e g r a l w i t h r e s p e c t t o t h ep a r a m e t e r s Y l , y 2 , . . . , y n . H o w e v e r , q u e s t i o n s i m m e d i a t e l y a r i s e a b o u t t h ele f t -inva r iance , r igh t - inva r iance a nd f in i t enes s o f such in teg ra ls . Fo r genera lt o p o l o g i c a l g r o u p s t h e s e b e c o m e p r o b l e m s i n m e a s u r e t h e o r y . U s i n g t h i s t h e -o ry Ha ar (1933) showed th a t fo r a ve ry l a rge c las s o f topo lo g ica l g roups ,wh ich inc ludes the l inea r L ie g roups , the re a lways ex i s t s a l e f t - inva r ian t in -t eg ra l and the re a lways ex i s t s a r igh t - inva r ian t in teg ra l . (Acco un ts o f thesed e v e l o p m e n t s , i n c l u d i n g p ro o f s o f t h e t h e o r e m s t h a t f ol lo w , m a y b e f o u n d i nt h e b o o k s o f n a l m o s ( 19 5 0 ), L o o m i s ( 1 9 53 ) a n d H e w i t t a n d R o s s ( 19 6 3 ). )

L e tf (T ) d iT - dy l . . . dyn f (T ( y l , . . . , Yn ) )az (y l , . . ., Yn)

1 n

a n d(3.13)

/ b l / a b nf ( T ) d~T - dy l . . , dyn f ( T ( y l , . . . , y n ) ) a~ ( y l , .. . , Yn) (3.14)1 n

be the l e f t - and r igh t - inva r ian t in teg ra l s o f a l inea r L ie g roup G , so tha tf = f I(T )d (3.15)J g J g

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L I E G R O U P S 45

f G f ( T T ' ) d ~ T = f f ~ f ( T ) d ~ T (3.16)f o r a n y T ~ E G a n d a n y f u n c t i o n f ( T ) fo r wh i ch t he i n t eg ra l s a re we l l de f i ned .H e r e a z ( y l , . . . , y ~ ) a n d a ~ ( y l , . . . , y n ) a r e l e f t - a n d r i g h t - i n v a r i a n t " w e i g h tf u n c t io n s " , w h i c h a re e a c h u n i q u e u p t o m u l t i p l ic a t i o n b y a r b i t r a r y c o n s t a n t s.T h e l e f t - a n d r i g h t - i n v a r i a n t i n t e g r a l s m a y b e s a i d t o b e f i n i t e if

a n d

~ ~ b l L b nd t T = - d y l . . . d y n a z ( Y l , . . . , Yn )

1 n

d ~ T - d y l . . . d y n a r (Yl, 999 Y n )1 n

a r e fi ni te . I f t h e m u l t i p l ic a t i v e c o n s t a n t s c a n b e c h o s e n s o t h a t a l ( Y l , . . . , Yn)a n d a t ( y 1 , . . . , y ~ ) a r e e q u a l , s o t h a t t h e i n t e g r a l s a re b o t h l e f t- a n d r ig h t -i n v a ri a n t, t h e n G is s a id t o b e " u n i m o d u l a r ' , a n d o n e m a y w r i te

d I T = d ~ T = d Ta n d

( 7 1 ( Y l , . . . , Y n ) - - 6 r r (Y l , . . . ,Yn ) - - o ' ( y x , . . . , y n ) .I f G h a s m o r e t h a n o n e c o n n e c t e d c o m p o n e n t , t h e i n t e g r a ls i n E q u a t i o n s ( 3.1 3)a n d ( 3 . 1 4 ) c a n b e g e n e r a l i z e d i n t h e o b v i o u s w a y t o i n c l u d e a s u m o v e r t h ec o m p o n e n t s .

T h e s ig n if ic a nc e o f t h e d i s ti n c t io n b e t w e e n c o m p a c t a n d n o n - c o m p a c t L ieg ro ups l ie s in t he f i r s t t wo o f t he fo l lowing t heo re m s , t he f i r st o f wh ich w aso r ig i n a ll y p r o v e d b y P e t e r a n d W e y l (1 92 7 ). T h e y i m p l y t h a t c o m p a c t L i eg r o u p s h a v e m a n y o f t h e p r o p e r t i e s o f f in i te g r o u p s , s u m m a t i o n o v e r a fi n it eg r o u p m e r e l y b e i n g r e p l a c e d b y a n i n v a r i a n t i n t e g r a l o v e r t h e c o m p a c t L i eg r o u p s , w h e r e a s f o r n o n - c o m p a c t g r o u p s t h e s i t u a t i o n i s c o m p l e t e l y d i f f e r e n t .T h e o r e m I I f G is a c o m p a c t L i e g r o u p , t h e n G is u n i m o d u l a r a n d t h ei n v a r i a n t i n t e g r a l

I ( T ) d T - d y e . . .

ex i s t s and i s f i n i t e f o r e v e r y c o n t i n u o u s f u n c t i o n f ( T ) . T h us a ( y l , . . . , Y n ) c a nb e c h o s e n s o t h a t

d T - d y l . . . d y n a ( y l , . . . , Y n ) ' - 1 .1 n

( A f u n c t i o n f ( T ) i s c o n t i n u o u s i f a n d o n l y i f f ( T ( y l , . . . , y n ) ) i s a con t i nuousf u n c ti o n o f y l , . . . , y n.)T h e o r e m I I I f ~ is a n o n - c o m p a c t L i e g r o u p t h e n t h e l e f t - a n d r i g h t -i n v a r i a n t i n t e g r a l s a r e b o t h i n f i n i t e .

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46 GROUP T HEOR Y IN PHYSICSFor non-co mp act g roups the ques t ion of when G is un im odu lar i s par t i a l ly

answered by the fol lowing theorem.T h e o r e m I I I I f G is A be l ian o r s emi- si mp le t hen G is un i modu l a r .

The def ini t ion of a semi-s imple Lie group is given in Chapter 11, Sect ion2 . T he o the r non-co mp act l inear L ie groups have to be inv es t iga ted ind iv idu-al ly . In pract ice, expl ici t express ions for weight funct ions are seldom needed.Indeed , in dea l ing wi th the compact L ie groups a l l tha t i s usua l ly r equi redis the knowledge (embodied in the f i r s t theorem above) that f ini te lef t - andr ight- invar iant integrals a lways exis t .