The Gerstenhaber structure on the Hochschild cohomology of a class of special biserial algebras

TL;DR

This paper characterizes the Gerstenhaber structure on the Hochschild cohomology of certain self-injective special biserial algebras, revealing a Lie algebra structure related to the Virasoro algebra and describing module decompositions.

Contribution

It determines the Gerstenhaber structure on Hochschild cohomology for a class of special biserial algebras, linking it to Virasoro algebra representations.

Findings

Hochschild cohomology in degree one is isomorphic to a sum of Virasoro algebra subquotients.

The Lie algebra structure in degree one is a direct sum of commuting Virasoro degree 0 copies.

Degree n cohomology decomposes into indecomposable modules over the Lie algebra.

Abstract

We determine the Gerstenhaber structure on the Hochschild cohomology ring of a class of self-injective special biserial algebras. Each of these algebras is presented as a quotient of the path algebra of a certain quiver. In degree one, we show that the cohomology is isomorphic, as a Lie algebra, to a direct sum of copies of a subquotient of the Virasoro algebra. These copies share Virasoro degree 0 and commute otherwise. Finally, we describe the cohomology in degree $n$ as a module over this Lie algebra by providing its decomposition as a direct sum of indecomposable modules.