Bracket structure on Hochschild cohomology of Koszul quiver algebras using homotopy liftings

TL;DR

This paper explores the Gerstenhaber algebra structure on Hochschild cohomology of Koszul quiver algebras using homotopy liftings, providing new insights and examples, including counterexamples to a conjecture.

Contribution

It introduces a novel approach to study the Gerstenhaber structure via homotopy liftings and applies it to specific quiver algebras, including counterexamples to a known conjecture.

Findings

Demonstrated the use of homotopy liftings in Hochschild cohomology

Provided examples of homotopy lifting maps for degree 1 and 2 cocycles

Identified Hochschild 2-cocycles satisfying the Maurer-Cartan equation

Abstract

We present the Gerstenhaber algebra structure on the Hochschild cohomology of Koszul algebras defined by quivers and relations using the idea of homotopy liftings. E.L. Green, G. Hartman, E.N. Marcos and O. Solberg provided a canonical way of constructing a minimal projective bimodule resolution of a Koszul quiver algebra. The resolution has a comultiplicative structure which we use to define homotopy lifting maps. We first present a short example that demonstrates the theory. We then study the Gerstenhaber algebra structure on Hochschild cohomology of a family of bound quiver algebras, some members of which are counterexamples to the Snashall-Solberg finite generation conjecture. We give examples of homotopy lifting maps for degree $2$ and degree $1$ cocycles and draw connections to derivation operators. As an application, we describe Hochschild 2-cocycles satisfying the Maurer-Cartan equation.