Batalin-Vilkovisky structure on Hochschild cohomology of zigzag algebra of type \(\widetilde{\mathbf{A}}_1\) (Q2164997)

This paper is a study of the Batalin Vilkovisky (BV) structure on the Hochschild cohomology of quantum algebras. First, the authors review the definitions of Hochschild homology and cohomology, cup product, Gerstenhaber bracket and BV algebra. Then the authors prove that the product of cohomology rings is essentially the connection of parallel paths. Finally the authors calculate the dimensions of the Hochschild groups. They also give explicitly the Batalin-Vilkovisky operator and the Gerstenhaber bracket on hochschild cohomology.