BV differential on Hochschild cohomology of Frobenius algebras

For a finite-dimensional Frobenius k-algebra R with a Nakayama automorphism ν  , we define an algebra HH⁎(R)ν↑. If the order of ν is not divisible by the characteristic of k, this algebra is isomorphic to the Hochschild cohomology algebra of R  . We prove that this algebra is a BV algebra. Moreover, we construct a BV differential on it in a canonical way. We use this fact to calculate the Gerstenhaber algebra structure and a BV structure on the Hochschild cohomology algebras of a family of self-injective algebras of tree type DnDn.