Batalin-Vilkovisky structure on Hochschild cohomology with coefficients in the dual algebra
Marco Armenta, Samuel Leblanc
TL;DR
The paper develops a BV-algebra structure on Hochschild cohomology with coefficients in the dual algebra $A^* = \mathrm{Hom}_k(A,k)$ by equipping $A^*$ with an associative $k$-algebra structure and ensuring a graded-commutative cup product on $H^\bullet(A,A^*)$, enabling the BV-operator $\bar{B}$ via Connes' differential. It extends this BV framework by introducing a bracket $[f,g]_{\psi}$ using a bimodule morphism $\psi: A^* \to A$, yielding a BV-algebra $(H^\bullet(A,A^*), \cup_*, [\,,\,]_*, \bar{B})$, and shows how this recovers known BV-structures in symmetric and Frobenius cases through explicit isomorphisms (e.g., via $Z: A \to A^*$). In symmetric algebras, $HH^\bullet(A)$ and $H^\bullet(A,A^*)$ BV-structures coincide under $Z$, while for Frobenius algebras with Nakayama automorphism $\mathfrak{N}$, the BV-structure on $H^\bullet(A,A_\mathfrak{N}^*)$ is isomorphic to $HH^\bullet(A)$ under suitable conditions, with an explicit operator $\Delta$ given by $Z_*\partial B_\mathfrak{N}^* \overline{\mathfrak{C}}$. The paper also proves that for monomial path algebras $A = kQ/\langle T\rangle$, $H^\bullet(A,A^*)$ is always a BV-algebra, illustrating broad applicability beyond the usual Hochschild cohomology with coefficients in $A$.
Abstract
We prove that Hochschild cohomology with coefficients in $A^*=\Hom_k(A,k)$ under conditions on the algebra structure of $A^*$ 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 $A = kQ/\langle T \rangle$, the Hochschild cohomology with coefficients in $A^*$ is always a Batalin-Vilkovisky algebra.