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Affine algebraic varieties affine algebraic groups
First Part Jordan decompositions unipotent
Quotients and solvable groups
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abelian affine algebraic group affine algebraic variety affine variety algebraically closed assume automorphism basis Borel subgroup Bruhat lemma Cartan subgroup char G characters Choose claim Clearly closed subgroup closed subset codimension commute conjugacy classes conjugate contains corollary cosets defined denote diagonalizable diagonalizable group dim G dimension elements of G endomorphism fact fibre finite dimensional fixed point follows immediately functions given GL(V group G Hence dim integral invariant irreducible components irreducible subset irregular unipotent elements Jordan decomposition k-algebra k-algebra homomorphism Let f Lie algebra line of type linear locally finite matrices maximal torus minimal morphism nilpotent noetherian open set orbit polynomial positive roots projective variety Proof proves the proposition quotient reductive group regular elements Remark root system semisimple elements semisimple group simple root simply connected subgroup of G subregular elements subspace subvariety surjective Symm unipotent elements unique vector space zero ZG(t ZG(x