Batalin-Vilkovisky structure on Hochschild cohomology of zigzag algebra of type $\widetilde{\mathbf{A}}_{1}$

TL;DR

This paper explicitly computes the Hochschild cohomology ring and Batalin-Vilkovisky structure of quantum zigzag algebras of type f0a1, providing detailed algebraic structures and operators.

Contribution

It introduces explicit calculations of Hochschild cohomology, the BV operator, and Gerstenhaber bracket for quantum zigzag algebras of type f0a1, advancing understanding of their algebraic structures.

Findings

Hochschild homology and cohomology dimensions are computed.

The Hochschild cohomology ring of A_q is explicitly determined.

The BV operator and Gerstenhaber bracket are explicitly described.

Abstract

In this paper, we study the Batalin-Vilkovisky structure on the Hochschild cohomology of quantum zigzag algebras $A_{q}$ of type $\widetilde{\mathbf{A}}_{1}$. We first calculate the dimensions of Hochschild homology groups and Hochschild cohomology groups of $A_{q}$. Based on these computations, we determine the Hochschild cohomology ring of $A_{q}$, and give the Batalin-Vilkovisky operator and the Gerstenhaber bracket on Hochschild cohomology ring of $A_{q}$ explicitly.