Construction of Miniversal Deformations of Lie Algebras

In this paper we consider deformations of finite or infinite dimensional Lie algebras over a field of characteristic 0. By “deformations of a Lie algebra” we mean the (affine algebraic) manifold of all Lie brackets. Consider the quotient of this variety by the action of the group GL. It is well-known (see [Hart]) that in the category of algebraic varieties the quotient by a group action does not always exist. Specifically, there is in general no universal deformation of a Lie algebra L with a commutative algebra base A with the property that for any other deformation of L with base B there exists a unique homomorphism f :B → A that induces an equivalent deformation. If such a homomorphism exists (but not unique), we call the deformation of L with base A versal.