Homotopy lifting maps on Hochschild cohomology and connections to deformation of algebras using reduction systems

TL;DR

This paper explores the structure of Hochschild cohomology for Koszul quiver algebras, introducing homotopy lifting maps and scalars, and connects these to algebra deformations via reduction systems.

Contribution

It provides a new description of homotopy lifting maps using comultiplicative structures and recurrence relations, linking Hochschild cohomology to algebra deformations.

Findings

Describes Gerstenhaber bracket via homotopy lifting maps.

Provides recurrence relations for scalars in homotopy lifting maps.

Connects Hochschild 2-cocycles to algebra deformations using reduction systems.

Abstract

We describe the Gerstenhaber bracket structure on Hochschild cohomology of Koszul quiver algebras in terms of homotopy lifting maps. There is a projective bimodule resolution of Koszul quiver algebras that admits a comultiplicative structure. Introducing new scalars, we describe homotopy lifting maps associated to Hochschild cocycles using the comultiplicative structure. We show that the scalars can be described by some recurrence relations and we give several examples where these scalars appear in the literature. In particular, for a member of a family of quiver algebras, we describe Hochschild 2-cocycles and their associated homotopy lifting maps and determine the Maurer-Cartan elements of the quiver algebra in two ways: (i) by the use of homotopy lifting maps and (ii) by the use of a combinatorial star product that arises from the deformation of algebras using reduction systems.