Construction of Miniversal Deformations of Lie Algebras

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 a Lie algebraLwith a commutative algebra baseAwith the property that for any other deformation ofLwith baseBthere exists a unique homomorphismf: A→Bthat induces an equivalent deformation. Thus one is led to seek aminiversaldeformation. For a miniversal deformation such a homomorphism exists, but is unique only at the first level. If we consider deformations with base spec A, whereAis a local algebra, then under some minor restrictions there exists a miniversal element. In this paper we give a construction of a miniversal deformation.