Symmetry And Group

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Extra resources for Group theory (Lie and other) (draft 2002)

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2. subgroups , Gr then: . i s u n i v e r s a l ( r e s p e c t i v e l y a d j o i n t ) i f and o n l y i f G, is. I n each c a s e i n ( b ) , t h e p r o d u c t i n ( a ) i s d i r e c t . Corollary 6: : SL2 i s normal i n Gi Since t h e i n t h i s case. i s a Chevalley group, G n c o r r e s p o n d i n g t o indecomposable components of each va Z If a . Hence t h e f u n d a m e n t a l w e i g h t s is g e n e r i c a l l y c y c l i c of o r d e r SLn ai l S i < , n - 1 , we have i a r e i n the l a t t i c e associated with t h i s representation.

YI=- [ -1 O 0 {+ 1 ) or or a PSC2 , . [H, X] = 2X v e c t o r space x -> V 9 i s isomorphic t o so that ker u C Exercise: {t 1) If Now 9 4 S i m p l i c i t y of 2)- has a s t 2 on a -:]as , Y -> XG with X -a , H -> a s t h e o r i g i n a l r e p r e s e n t a t i o n of Ha x. e x i s t s and CQ by C o r o l l a r y 5. i s u n i v e r s a l , each G H and SL2 i s u n i v e r s a l , t h e r e q u i r e d homomorphism Since has ' .

P r o v i n g ( c 1. Then: h (t)o-l= h g (a) C1 p r o d u c t of ( t )= a n e x p r e s s i o n a s a wup hvs , independent of t h e r e p r e s e n t a t i o n space. (b) oC! x R ( t ) w-1c = X, O ( ~ tw )i t h , c as in a- Lemma 13(a). (c) Proof: h c ( t ) x R ( u ) h a ( t )-1 = Y, (t u ) To p r o v e ( a ) we a p p l y b o t h s i d e s t o -I Q hl (t)iuo u ? $,a> Lemma 2 0 : Now and i s independent ~ h i c h ~ p r o v e( sa ) . wa(t)-' Note t h a t X . 1 , c(a,p) = c(c,-0) By ( b ) and (?

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