## 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.

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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.