Extensions of Algebraic Groups

Details Extensions of Algebraic Groups Extensions of Algebraic Groups

2004-02-27

arxiv.org/abs/math/0402453v1

Mathematics

Algebraic Geometry

Scientific paper

Let $G$ be a connected complex algebraic group and $A$ a connected abelian algebraic group endowed with an algebraic action of $G$ by group automorphisms. In the present note we describe the abelian group $\Ext_{alg}(G,A)$ of algebraic group extensions of $G$ by $A$ in terms of a short exact sequence relating the ext-group to a relative second Lie algebra cohomology space and the fundamental group of the commutator group. Our second main result is an analog of the Van-Est Theorem for algebraic group cohomology, saying that for an algebraic $G$ module $\a$ and $p \geq 0$ the algebraic group cohomology $H^p_{alg}(G,\a)$ is given by the relative cohomology of the Lie algebra $\g$ with respect to the Lie algebra of a maximal reductive subgroup.

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