Batalin-Vilkovisky structure on Hochschild cohomology of zigzag algebra of type ${\tilde{\mathbf{A}}}_1$

Abstract

In this paper, we study the Batalin-Vilkovisky structure on the Hochschild cohomology of quantum zigzag algebras $A_q$ of type ${\tilde{\mathbf{A}}}_1$. 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.