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

Abstract: In this paper, we study the Batalin-Vilkovisky structure on the Hochschild cohomology of quantum zigzag algebras

A_{q}

of type

w i d e t i l d e {m a t h b f 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.

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