Extensions of algebraic groups

Abstract

Let G be a connected complex algebraic group and A an abelian connected algebraic group, together with an algebraic action of G on A via group automorphisms. The aim of this article is to study the group of isomorphism classes of extensions of G by A in the algebraic group category. We describe this group as a direct sum of the group Hom(π1([G,G]), A) and a relative Lie algebra cohomology space. We also prove a version of Van Est’s theorem for algebraic groups, identifying the cohomology of G with values in a G-module a in terms of relative Lie algebra cohomology.