Symmetry And Group

## Infinitesimally Central Extensions of Chevalley Groups by W. L. J. van der Kallen

Posted On March 23, 2017 at 10:32 am by / Comments Off on Infinitesimally Central Extensions of Chevalley Groups by W. L. J. van der Kallen

By W. L. J. van der Kallen

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The contents of this publication were used in classes given through the writer. the 1st was once a one-semester path for seniors on the collage of British Columbia; it used to be transparent that reliable undergraduates have been completely able to dealing with simple workforce idea and its program to easy quantum chemical difficulties.

Additional resources for Infinitesimally Central Extensions of Chevalley Groups

Example text

We don't need this method. (See section 10). §7. The e x t e n s i o n ~ : G* ~ G. 3)). In this section we make some remarks about such a homomorphism. We suppose that the c o d o m a i n of ~ is an almost simple C h e v a l l e y group G, h a v i n g K as its Lie algebra. Let G * denote the d o m a i n of ~. If # is such that d~ is a u n i v e r s a l central e x t e n s i o n of ~, then the r e s t r i c t i o n of ~ to the c o n n e c t e d component of G * also has that property. is connected.

8, Table from the above: K (see [7 ], cf. 1). to a C h e v a l l e y so we can apply subalgebras, 1). K the following two are equivalent: (i) To find a h o m o m o r p h i s m (ii) To find an algebraic algebra, ~ such that group d~ = ~. G* w h i c h has K* as its Lie such that the [ p ] - s t r u c t u r e on K" is invariant under Ad. REMARK. 8. first morphism tesimally central Consider sition algebras solution. cussed d~ is a central Then extension ~ is called an infini- of G. e. h = [h, h]). 2), there 13 (see T h e o r e m from Propo- homomorphism of Lie p is a u n i v e r s a l of finding cen- a homomor- is the p r o b l e m of finding X such that d X = p.

If A. is of type a, t h e n L. 1 lie in one Obviously, the orbit and have there is at m o s t classification of type Note that 6 i is a short case 1. Z is of t y p e In the (see roots [26], ducible Table have root II). sums 1, we (see its w e i g h t s Corollary b in A. 14). 12). F Comparing ~ 0 Fr in Li, 2. Z is of type in case one A. of type l multiplicity representation case 1 (see , p = 2. 4 representation ~ of G, w i t h of the r e p r e s e n t a t i o n As because b. irreducible short multiplicity of d e g e n e r a t e case 6i is i r r e d u c i b l e i see t h a t the (cf.