Batalin-Vilkovisky structure on Hochschild cohomology with coefficients in the dual algebra

Abstract

We prove that Hochschild cohomology with coefficients in $A^*=\Hom_k(A,k)$ under conditions on the algebra structure of $A^{\ast }$ is a Batalin-Vilkovisky algebra. We also show that for symmetric and Frobenius algebras, this recovers the known BV-structures in Hochschild cohomology with coefficients in $A$ but admits an easy-to-describe BV-operator. Finally, we show that for monomial algebras $A=kQ/⟨T⟩$, the Hochschild cohomology with coefficients in $A^{\ast }$ is always a Batalin-Vilkovisky algebra.

13 pages