BV-differential on Hochschild cohomology of Frobenius algebras

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For a finite-dimensional Frobenius k-algebra R with the 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. We use this fact to calculate the Gerstenhaber algebra structure and BV-structure on the Hochschild cohomology algebras of a family of self-injective algebras of tree type Dₙ.