By W. L. J. van der Kallen

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**Example text**

Almost (use over simple Over is Chevalley group as G. 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.

8, Table 1). So r e l a t i o n s y @ 0. 7. (0), (12) imply (ker w) (Define H* in the same way). 14). 15), isomorphic to gR (Use (0)). So and w is an isomorphism. 14. COROLLARY. 2)). (i) ~k is centrally closed if and only if there is no d e g e n e r a t e sum. (ii) For each d e g e n e r a t e sum~ its m u l t i p l i c i t y in ~* i__ss1. generate. (iv) ~. If root lensths are e~ual, then (ker ~)0 = 0. b. If ~ is of type F 4 and p = 2, then dim (ker ~)0 = 2. c. If ~ is of type B I and p = 2, then dim (ker ~)0 = 1.

8. PROOF CONTINUED. 14). ' ~ g2z There is a h o m o m o r p h i s m T: g ~ such that r 0 Y = ~. The central trick proves So there can't be more relations between the H~, then there are f between the H a in gz~ " This proves: (18) The subspace (g~)0' generated by the H a, has (1) and (3) as d e f i n i n g relations. The other components of the grading of g %s one generator. ~ It is a is free. If 6 is d e g e n e r a t e with respect to n, then ( g ~ ) 6 has Z~ as generator, (see (12)). )6 is either zero or n-cyclic.