Gerstenhaber algebra structure on the Hochschild cohomology of quadratic string algebras

TL;DR

This paper investigates the Gerstenhaber algebra structure on Hochschild cohomology for quadratic string algebras, providing explicit computations, generators, and conditions for non-trivial algebraic structures.

Contribution

It introduces a method to compute Hochschild cohomology and describe its Gerstenhaber algebra structure for quadratic string algebras, including explicit formulas and conditions.

Findings

Computed Hochschild cohomology groups using Bardzell's resolution

Derived formulas for cup product and Lie bracket

Identified conditions for non-trivial Gerstenhaber structures

Abstract

We describe the Gerstenhaber algebra structure on the Hochschild cohomology HH*$(A)$ when $A$ is a quadratic string algebra. First we compute the Hochschild cohomology groups using Barzdell's resolution and we describe generators of these groups. Then we construct comparison morphisms between the bar resolution and Bardzell's resolution in order to get formulae for the cup product and the Lie bracket. We find conditions on the bound quiver associated to string algebras in order to get non-trivial structures.