By Wehrfritz B.A.F.
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Extra info for 2-Generator conditions in linear groups
We r e p e a t the d e f i n i t i o n : I" f = f o T g g for f ~ CX. 1. i. 1. Z. and X' be manifolds, functions. CX and C X ' Show that an arbitrary is a h o m o m o r p h i s m Exercise 4. i. 3. ring homo- of ]R-algebras. L e t the s i t u a t i o n be as in e x e r c i s e (i = 1, 2) be m a p s the corres- s u c h t h a t ~1 = ~Z" 4. 1 a n d Show t h a t t h e n ~1 = Og" Let the situation be as in exercise 4. I. i and consider the map [x, x'] from maps @~ - . > ~ X > X' to r i n g h o m o m o r p h i s m s Exercise for p a r a c o m p a c t > [cx', cx] > CX d e f i n e d by 4.
Lg defines A. (i = 1, . - - , lg vectorfields n) f o r m a base -49 - of T X for all g C G . g COROLLARY Example Then LIR ~ 4. 3. 6. 4. 3. 7. T h e manifold of a Lie group G is parallelizable. Consider IR with its additive Lie group structure. IR as vectorspace, because the tangent space of IR at O is ]R . T h e r e is only one possible Lie algebra structure on 11% , defined by = Ofor c B y the s a m e Now L 11~ = IR for the additive group that considering endomorphiams map argument, let V be n-dimensional first r e m a r k g G G.
Of t h e b a s e o f V c o r r e s p o n d t o t w o isomorphisms which differ by an inner automorphism Example 3. 1. 7 Let G e of G a t t h e i d e n t i t y GL(V) of G L ( n , G be a Lie group and with its Lie group structure (example e of G 3. 1. 3). and its natural assigning of L i e g r o u p s . to each tangent vector Lie groups. The natural its origin, two choices > G L ( n , JR) ]1%). TG the tangent bundle Consider the tangent injection If G e i s e q u i p p e d w i t h t h e L i e g r o u p s t r u c t u r e a homomorphism Because j : G space >TG.