Construction of Miniversal Deformations of Lie Algebras

Abstract

We consider deformations of finite or infinite dimensional Lie algebras over a field of characteristic 0. There is substantial confusion in the literature if one tries to describe all the non-equivalent deformations of a given Lie algebra. It is known that there is in general no "universal" deformation of the 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: A -> B that induces an equivalent deformation. Thus one is led to seek a "miniversal" deformation. For a miniversal deformation such a homomorphism exists, but is unique only at the first level. If we consider deformations with base spec A, where A is a local algebra, then under some minor restrictions there exists a miniversal element. In this paper we give a construction of a miniversal deformation.