BV-operators and the secondary Hochschild complex

We introduce the notion of a BV-operator $\Delta =\lbrace \Delta ^n:V^n\longrightarrow V^{n-1}\rbrace _{n\ge 0}$ on a homotopy $G$-algebra $V^\bullet $ such that the Gerstenhaber bracket on $H(V^\bullet )$ is determined by $\Delta $ in a manner similar to the BV-formalism. As an application, we produce a BV-operator on the cochain complex defining the secondary Hochschild cohomology of a symmetric algebra $A$ over a commutative algebra $B$. In this case, we also show that the operator $\Delta ^\bullet $ corresponds to Connes’ operator