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.
Categories: math.RA math.KT math.RT