From Lie Algebras of Vector Fields to Algebraic Group Actions

From the action of an affine algebraic group G on an algebraic variety V , one can construct a representation of its Lie algebra L(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 → DerK(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.