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abelian group abelian variety adjoint algebraic group analytic representation analytic subgroup analytic vector arbitrary assume base point characteristic multipliers coefficients commutative commutative diagram compact subgroup completes the proof complex analytic group complex analytic representation consider contained Corollary corresponding defined denote diagram differentiable dimension direct sum divisor dual duality map element equation equivalent exists extension faithful complex analytic field of definition finite number fixed follows function given Hence homomorphism homotopy implies induces integer invariant isomorphism Lemma Let G Lie algebra Lie group linear variety manifold marker Math matrix maximal metric space module nilpotent non-degenerate notation obtain operation p-free periodic solution polynomial positive pre-norm proof of Theorem Proposition prove r-finite reductive regular result satisfying sequence space submodule subset subspace Suppose symmetric Theorem Thue system tion topological unique vanish word problem zero