BV-operators and secondary Hochschild complex

We introduce the notion of a BV-operator $\mathrm{\Delta }=\{{\mathrm{\Delta }}^n:V^n⟶V^{n-1}{\}}_{n\ge 0}$ on a homotopy $G$-algebra $V^{\bullet }$ such that the Gerstenhaber bracket on $H(V^{\bullet })$ is determined by $\mathrm{\Delta }$ 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 $B$.