Examples of infinity and Lie algebras and their versal deformations

This article explores some simple examples of L-infinity algebras and the construction of miniversal deformations of these structures. Among other things, it is shown that there are two families of nonequivalent L-infinity structures on a 1|1 dimensional vector space, two of which are Lie algebra structures. The main purpose of this work is to provide a simple effective procedure for constructing miniversal deformations, using the examples to illustrate the general technique. The same method can be applied directly to construct versal deformations of Lie algebras.