In the papers [La 5,6], we introduced the notion of (non-commutative) phase algebras (space) Phn(A), n = 0, 1, ...,∞ associated to any associative algebra A (space), defined over a field k. The purpose of this paper is to prove that this construction is useful in non-commutative deformation theory for the construction of the versal family of families of modules, see [La 4]. In particular we obt...

We describe a canonical L∞ structure on the total complex of a semicosimplicial differential graded Lie algebra and give an explicit descriprion of the Maurer-Cartan elements and of the associated deformation functor in the particular case of semicosimplicial Lie algebras. We use these results to investigate the deformation functor associated to a sheaf of Lie algebras L and to show that it is ...

Deformations of differential complexes occur as a part of deformation theories of algebraic structures, but are interesting in their own right. In fact, this theory enjoys many properties that are ‘desirable’ but rare in other theories. We show (a) that the moduli space of deformations of a differential can be classified by a purely combinatorial object, namely, by monotone sequences in a count...

An algebraic deformation theory of dialgebra morphisms is obtained.

Deformation theory requires solving Maurer-Cartan equation (MCE) associated to an DGLA (L-infinity algebra). The universal solution of [HS] is obtained iteratively, as a fixed point of a contraction, analogous to the Picard method. The role of the Kuranishi functor in this construction is emphasized. The parallel with Lie theory suggests that deformation theory is a higher “dimensional” version...

In this paper we describe the deformation theory of sandwiched sin-gularities in terms of-constant deformations of plane curves, and a divisor of points on it. This leads to an immediate understanding of the smoothings of sandwiched singularities in terms of pictures: certain conngurations of points and curves with only d-fold points in the plane. The topology of a smoothing can be described co...