Examples of miniversal deformations of infinity algebras

Abstract

A classical problem in algebraic deformation theory is whether an infinitesimal deformation can be extended to a formal deformation. The answer to this question is usually given in terms of Massey powers. If all Massey powers of the cohomology class determined by the infinitesimal deformation vanish, then the de- formation extends to a formal one. We consider another approach to this problem, by constructing a miniversal deformation of the algebra. One advantage of this approach is that it answers not only the question of existence, but gives a construction of an extension as well. In this paper, we study some examples of miniversal deformations of infinity algebras, and use these examples to illustrate how to use a miniversal deformation to determine when an infinitesimal deforma- tion extends to a formal deformation. Actually, using a miniversal deformation, one can construct such an extension explicitly. Also, the obstruction to an extension can be computed by this method. An infinitesimal deformation extends to a formal one precisely when the unique morphism from the base of the universal infinitesimal defor- mation to the base of the given deformation inducing the infinitesimal deformation can be lifted to a morphism from the miniversal defor- mation to the formal power series ring in the parameter of the defor- mation. A nice property of this algebraic approach is that it answers more than the question of existence; in fact, it gives a construction of an extension of the infinitesimal deformation to a formal deformation. Moreover, since the question is reduced to studying the morphisms of the base of the miniversal deformation to a formal power series ring, the