From Lie algebras of vector fields to algebraic group actions

Abstract

The action of an affine algebraic group G on an algebraic variety V can be differentiated to a representation of the Lie algebra L(G) of G by derivations on the sheaf of regular functions on V . Conversely, if one has a finite-dimensional Lie algebra L and a homomorphism ρ : L → Der_K(K[U]) for an affine algebraic variety U, one may wonder whether it comes from an algebraic group action on U or on a variety V containing U as an open subset. In this paper, we prove two results on this integration problem. First, if L acts faithfully and locally finitely on K[U], then it can be embedded in L(G), for some affine algebraic group G acting on U, in such a way that the representation of L(G) corresponding to that action restricts to ρ on L. In the second theorem, we assume from the start that L = L(G) for some connected affine algebraic group G and show that some technical but necessary conditions on ρ allow us to integrate ρ to an action of G on an algebraic variety V containing U as an open dense subset. In the interesting cases where L is nilpotent or semisimple, there is a natural choice for G, and our technical conditions take a more appealing form.