7/30/2019 Lie Algebras and Lie Groups

1/4

ALie A lgebras and L ie Groups" Bas ic Not ions

A L i e a l g e b r a A i s a l i ne a r spa c e ove r some f i e ld F , sa y , f o r in s t a nc e , t hef ie ld of r e al n u m b e r s R o r o f c o m p l e x n u m b e r s C , t o g e t h e r w i t h a b i n a r yo p e r a t i o n [.,-] , w h i c h m a p s p a i r o f e l e l n e n t s i n A i n to a l l e l e lne n t in A:

[ ., . ] : ( h , . 9 ) c A A , [h, .q] E .4. (A .1 )

I t i s c a l l e d the L i e b r a c k e t a n d s a t i s f i e s t h e f o l l o w i n g p r o p e r t i e s :9 b i l i n e a r i t y :[ ~ h + V k , ~ ] = ~ [ h , ~ ] + V [ k , ~ ] , [ h , ~ g + v k ] = ~ [ h , g ] + b i b , k ] , (A .2 )

for a ll a , b C F an d h , k ,g E A;9 a n t i c o m m u t a t i v i t y : [ h , . q ] = - [ . q , h i ,for a l l h, g r A;9 J a c o b i i d e n t i t y :

(A .3 )

[ h , [ k , g ] ] + [ k , [ 9 , h ] l + [ ~ , [ h , k ] = o . (A .4 )for a l l h, k, g E A.

I t i s e v i d e n t t h a t i n g e n e r a l t h e L i e b r a c k e t i s n o t a s s o c i a t i v e , i . e .[ h , [ k , g ] ] r [ [ h , k ] , g ] , (A .5 )

t h e a s s o c i a t i v e l a w b e i n g r e p l a c e d b y t h e J a c o b i i d e n t i t y .T h e d i m e n s i o n o f A is , o f c o u rs e , d e t e r m i n e d b y t h e c a r d i n a l i t y o f t h e b a s i s

se t { h j } j , w h i c h s p a n s t h e u n d e r l y i n g l i n e a r s p a c e .A s a c o n c r e t e e x a m p l e o f a L ie a lg e b r a , w e m a y m e n t i o n t h a t t h e E u c l i d e a n

s p a c e R 3 b e c o m e s a t h r e e - d i m e n s i o n a l L i e a l g e b r a w i t h t h e L ie b r a c k e t t a k e na s t h e c r o s s p r o d u c t o f v e c t o r s . A l s o , a n y a s s o c i a t i v e a l g e b r a , w i t h a s s o c i a t i v ep r o d u c t f g ( h e re s i m p l y d e n o t e d b y j u s t a p o s i t i o n ) , b e c o m e s a L i e a l g e b r au n d e r t h e d e f i n it i o n o f t h e L i e b r a c k e t a s t h e c o m m u t a t o r , a l so c a l l e d L i ep r o d u c t , o f e l e m e n t s i n t h e a l g e b r a :

{ h , g ] = h ~ - g h . ( A . 6 )

7/30/2019 Lie Algebras and Lie Groups

2/4

520 L i n e a r R a y a n d W a v e O p t i c s n P h a s e 5 p a c e

I n a s e n se t h e c o m n m t a t o r is a m e a s u r e o f h o w n o n - c o m m u t a t i v e t h e a lg e b r ais . T h e a l g e b r a o f i n v e r t i b l e r e a l s q u a r e m a t r i c e s is m a d e i n t o a L i e a l g e b r au n d e r th e. L i e t ) r o d u ( ' t ( A .6 ) , i n v o l v i n g t h e f a m i l i a r r o w- b y - c o l ~m n l m u l t i p l i -c a t i o n o f m a . t ri c c s. Be s i ( te s , i n t h e t e x t i t is s e e n t h a t t h e a l g e b r a o f t h e 2 Di l lhomogelu;O~ lS l ine ar t )oly non fia ls (wi t l l rest )cot to a ,( tr a .cqui rcs a Liea . lget ) ra s tn u : t ur c lul ( te r ti l t; P oisso l l bra ,(:ket o t)era t io~l , as s in l i la r ly ( toes th ea l g e l ) r a ~ f 2 1 ) l~on~ogelm~n~ s (t~a,r l)~)lyll~)~fials.

A h,o mo mo ' t7~ h , i s m a 1)(:t,w(:(:ll l,w() l.i e a,lg;el~ra,s A m u l / 3 ()v(:r t,l~(: sa n le ti(:l(t/~':a : A , B ( A . 7 )

(letixu~s a, li~('a,r ~ lm,l)t>i~g ~ f A i~t() B , s~u:t~ tlm,t tl~e I.ie Iwa,t 'ket i~ B ~ f illm,ge(telellltU ltS ()f ~4 is ta,ktu~ I,() tl w illm,g('. ()f til e l.i e lwa~ :ket ill A ()f 1,11e ()rigina,1(~lelliell|,S; l~m~l('.ly:

[ ~ ( h ) , ~ ( g ) l = ~ ( [ h , g ] ) v h , g e , 4 . ( A . S )If slu :ll a ll()l,,()l,l(~I't)llisll, is ()ll('~-t()-()ll(; (()r, . f i t i t /r s() tlmt tll(' ilw(;rs(; l~m.t)-t)illg exist,s, it, is

7/30/2019 Lie Algebras and Lie Groups

3/4

Appendix 52 1A l s o , a L i e a l g e b ra A i s s imple i f i t c o n t a i n s n o n o n z e r o i d e a ls , w h i l s t i t issemi - s imp le i f i t c o n t a i n s n o n o n z e r o a b e l i a n i d e a ls . A s i m p l e L i e a l g e b r a i s

s e m i - s im p l e . I t t u r n s o u t t h a t s e m i - s i m p l e L ie a l g e b r a s a r e t h e d i re c t s u m o fs i m p l e ( n o n a b e l i a n ) L i e a l g e b r a s .A L i e g r o u p ~ c o m b i n e s t h e c o n t i n u o u s s t r u c t u r e o f a d if f e r e n ti a b l e m a n i f o l dw i t h t h e b a s i c p r o p e r t i e s o f g r o u p s . A g r o u p i s a s e t o f e l e i n e n t s i n w h i c h i tis d e f in e d a b i n a r y o p e r a t i o n ( t h e grou p mu l tip l ica t ion ) , w h i c h m a p s a n y t w oe l e m e n t s i n G i n t o a n c l e m e n t i n G a s w e ll , a n d s a t is f i e s t h e p r o p e r t i e s :

9 associativity: H . ( G . K ) = ( H . G ) . K (A. 13)for a l l H , G, K E {7;9 existence of identity:

3 I E G 9 I . H - H ( A .1 4 )for allH EG;9 existence of unique inverse:

3 H -1 E ~} 9 H - 1 - H - H - H - 1 - I . ( A .1 5)f l ) r every H C G.

A f a m i l ia r e x a m p l e is t h e g r o u p 7 ),. o f a ll p e r m u t a t i o n s o f n o b j e c t s , w h i c hc o n s i s t s o f n ! o p e r a t i o n s . A l s o , t h e c o l l e c t i o n o f r o t a t i o n s o f t h e c i r c le i n ap l a n e t h ro u g h m u l t i p l e s o f a g i v e n a n g l e tg, s a y t9 - 2~ w i t h r e s p e c t t o a f i x e dn ~p o i n t , c o n s t i t u t e s a g r o u p I n a d e of n d i s t i n c t o p e r a t i o n s .

I n o r d e r t h a t G b e a L i e g r o u p , i t s g r o u p o p e r a t i o n m u s t b e c o n s i s t e n t w i t ht h e c o n t i n u o u s s t r u c t u r e o f t h e u n d e r l y i n g m a n i f o l d , a n d s o i t m u s t d i f fc r cn -t i a b l e a s w e ll . T h i s n a '/ v e ly m e a n s t h a t i f H a n d G i n G m u l t i p l y t o W - H - Gin ~ , t h e n t h e g r o u p o p e r a t i o n t a k e s a n y e l e m e n t s J a n d K , r e s p e c ti v e ly d r a w nf r o m a n e i g h b o r h o o d o f H a n d a n e i g h b o r h o o d o f G , t o t h e g r o u p e l e m e n t Yin a n e i g h b o r h o o d o f W .A s a si m p l e e x a m p l e o f L i e g r o u p s , w e m a y m e n t i o n t h a t t h e s e t o f r e a ln u m b e r s , e x c l u d i n g 07 f o r m s a L ie g r o u p w i t h r e s p e c t t o t h e f a m i l ia r m u l t i p li -c a t i o n o f n u m b e r s . T h e s e t o f t r a n s l a t i o n s o f a s t r a i g h t l i ne w i t h r e s p e c t t o ar e f e r e n c e p o i n t a l so c o n s t i t u t e s a L ie g r o u p , e a c h e l e m e n t o f w h i c h b e i n g i n d i -v i d u a l i z a b l e ( i . e . p a r a m e t e r i z a b l e ) b y a r e a l n u m b e r ( w h i c h s p e c i f i e s t h e s h i f tf r o m t h e i d e n t i t y ) . A l so , t h e e n s e m b l e o f r o t a t i o n s o f t h e c i r cl e w i t h r e s p e c t t oa f i xe d p o i n t b y a n a n g l e ~9, a l l o w e d t o r a n g e c o n t i n u o u s l y t h r o u g h t h e i n t e r v a l0

7/30/2019 Lie Algebras and Lie Groups

4/4

5 2 2 L inear Ray and W ave Opt i c s in Phase Spac e

c a n b e ( t if f e r en t l y i n t e r p r et e ( 1 a ( : c o r d i n g t o t h e a d o p t e d ( t ec ( )m t ) o si t io n . A n -o t h e r e x a m p l e is t h e g r o u p G L ( ~ t ) o f n x n i n v e r t i b l e m a t r i c e s , o r tl~ e r e l e v a n tsut )gro~H) O ( n ) o f o r t h o g ( m a l m a t r i c e s .

1R.()~ghly s t ) ca k i n g , L i e a . l ge t) r a s ex t ) ( m en t i a t e t o L i e g r (nq ) s . M ( ) r e t ) r e ( : i s e l y ,t h e t a n g e ~ t s t) a< ' e a t t h e i ( h m t i t y o f a L i e g r( n~ t ) i s a l w a y s a L i e a l g e t ) r a . T h i s L i ea l ge l ) r a (h ; t e i ' n f i nes t he h ) (: al s t r ~ ( : t ~ r ( ; ( )f t he L i e g ro~ l t) t h r (n~ gh t he ex t ) ( )n en t i a lnm.I). (~,()~si(t( ' .ri~g, f()r i l l~ st ra ti v e tn~rI)()s(;s, a. ()n(;-t) aI 'a.~ u;ter I, i(; gr (n q ) {7,S) ~ )l )( )s(: tlm.l, ( ; ( t ) r(;l)r(:selll,s a sni()()l,h (:lit 'v(: in ~ ; l )assi l~g ( , l~r()~gh ( ,h(: groupi(le~Itity, (: .g. I -: (;(()). F()r va hu :s bt ()f ( ,l~(: r ( :al l )a ra~ (:( , ( : r t i l~ l i~dl , ( :s imally(:h)s(; t() (), (,lle (:()rr(:sl)()~uli~g (:l(:~n(;~t ( ; ( b t ) iI~ ~ (:()~(:s t() l)(: i~fi~fi(,(:simally( : ] ( ) s e (, ( ) t l ~ ( : i ( ] ( : ~ t i t y I = ( 1 ( ( ) ) , i . ( : .

( A . 1 6 )tllllS (;St,al f l is l l i l lg a ~l lal) l ) i l lg ( ) f ( ; l ( ; 1 1 1 ( ; l l l , S (; ' (()) ill tll(' l,i( ', alg('.l)ra t() tt~(; i~-ti~fitesi~ m.l ge ~ mr at( )rs G((S t) ()f (,l~(; Li e gr()~q). It ix (;vi(h ;~ t t lm.t l ,l~(; sl, r~u:t,~r(;()f tt~(', a lg(; 1)r a ix r(;fi(;('l.(:(! l() (' al ly i~(,() tlm.(, ()f l,l~(; gr () ~ l) i~ a ~ (;ig lfl)( )rh( )()(t ()ft l~( ; i ( le~t i ty.

S~ q)t)()s(; ~l()w (,() l, ak(; a. fi~ ii,e ~nn ~d)(;r t ~ IR. () f ('.()~n's('., ()~(; ('a ~ g() i~ fi ni te s-i~ m,l ly (:l()s(; 1,()(), 1)y l)r()l )(,rly f ra( 'l,i ()~ i~ g t,; s(;(, r()~ gl~ ly, bl - a~ (l le t 1,1~(;I I I .in(,(:g(:r ')) )g( ) (,() infilfi(,y: '))) --+ c~ . Sill(:(:, 1)y vir(,~i(: ()f (,lle ('l()s~n'(: l)r ()l )e rt y ()ftl~(: gr()~ q), f()r a ~ y i~(,(:g(:r '))),, [(7'(bt)]'" l)(:l()~ gs (,o (,l~(: gr ()~ l), w(: fi~m.lly lm.ve

t (7,' '" to, ' (())[ ( ; (~t ) ] ' " ,.~ [ I + -- _ (())] ~ ( ' , (1 .1 7 )I l l ,/ , lnls esl, at ) l is l l i l lg al l ( :xt)()Imll t ial nmt)t)i21g l, l r (m gll ( , ll ( ; rea l lm.rmllel ,er t ( )feh:nl( :nl ,s G '( 0 ) in ( ,he Li( : alg(: l )r a ( , ()" f ini l, (:" eh:nle nl , s ()f l ,h(: I ,i ( : gr(n q ) .

H isto ri( : al l y, lS e a lg el )r as lm.ve en ler ge (t | )y ( :oIlsi (h:r i l lg ( :l( :IIl( :l ll , s in L iegTout ) s i n f i n i t e s i m a l l y ( :h )se t () t h e g r ( ) l q ) i ( t en t i ty .

D u e t o t h i s i n t i m a t e l i n k l ) e t w e e n L i c a l g c | ) r a s a .n (1 I , ie g r o u p s (,lu: ( : h a ra c -t e r i za . t i on ( )f L i c a l g eb ra s ~ llent, i one ( t a , t) ()ve r e f l ec t s i n an an a l o g( ms ( : ha ra , c te r -i z a t i o n ( o r , n l o r e p r e c i s e l y , c l a s s i f i c a t i o n ) o f L i e g r o u p s .

A r e p r e s e i l t a t i o n o f a L i e g r o u p {~ o n a v e c t o r s pa,(:(; V i s a h o m o m o r p h i s mo f {~ i n t o t h e g r o u p o f a u t o m o r p h i s m o f V ( i. e. o f t r a n s f o r m a , t i o n o f V in t oi t s e l f ). I f a tin .s is s e t is c h o s e n f o r t h e v e c t o r s p a c e V , t h e r e p r e s e n t a t i o nc a n e q u a l l y t )e i n t e n d e ( t a s a. h o m o m o r p h i s m o f {7 t o t h e g r o u p G L ( n ) , a n da c c o r d i n g l y i t t ) r o v id e s a m a t r i x r e p r e s e n t a t i o n o f t h e g r o u p . I n a s e n s e, ag , o u p r e t ) r e s e n t a t i o n I n a y y i e h t a d e s c r i p t i o n o f a g r o u p i n t e r m s o f m o r e" v i s i b l e " o t ) j e c t s .