Affine algebraic group

Condiary. Every affine algebraic group is isomorphic to an algebraic matrix group. 

Theorem. Let G be an affine algebraic group scheme over a field. Assume G is smooth and connected, and let H be a proper closed subgroup. Then dim H < dim G. 

Theorem. Let kbe a field of characteristic zero. Let G be a connected affine algebraic group schemeacting linearly on V. A subspace W of Vis stable under G iff it is stable under Lie(G). 

Corollary. Let k be a perfect field. Let F be an affine algebraic group scheme over k which is isomorphic to G0 over k. Then actually F G . 

Corollary. Let F - G be a quotient map of affine algebraic group schemes, and assume the kernel N is smooth. Then F(k,)- G(kt) is surjective, and there is an exact sequence... 

Theorem. (Lang) Let k be a finite field, and G an affine algebraic group scheme which is connected. Then Hl(k/k, G) is trivial. 

Hochschild, G. Introduction to Affine Algebraic Groups (San Francisco Holden-Day, 1971). Mainly algebraic matrix groups, with Hopf-algebraic treatment. The emphasis is on characteristic zero and relation with Lie algebras. 

Action of an affine group scheme 21 Adjoint representation of G 100 Affine algebraic group 29 Affine group scheme 5 Algebra 3... 

Let G be an affine algebraic group scheme. Show that always dim Lie(G) > dim G. [Pass to k and note Lie(Gred) S Lie(G). A ring-theoretic proof is also possible 1. 

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