The Hochschild cohomology ring of a Frobenius algebra with semisimple Nakayama automorphism is a Batalin-Vilkovisky algebra.
Let \(A\) be an associative algebra over a field \(k\). The Hochschild cohomology of \(A\), \(HH^*(A,A)\), has a Gerstenhaber structure [\textit{M. Gerstenhaber}, Ann. Math. (2) 78, 267-288 (1963; Zbl 0131.27302)]. In [\textit{T. Tradler}, Ann. Inst. Fourier 58, No. 7, 2351-2379 (2008; Zbl 1218.16004)], it was shown that for finite dimensional symmetric algebras this cohomology has a richer structure, \(HH^*(A,A)\) is a Batalin-Vilkovisky algebra. This was generalized to Calabi-Yau algebras by \textit{V. Ginzburg} [Calabi-Yau algebras. \url{arXiv:math/0612139}] and to twisted Calabi-Yau with certain condition by \textit{N. Kowalzig} and \textit{U. Krähmer} [J. Reine Angew. Math. 697, 159-219 (2014; Zbl 1352.16013)]. The aim of this paper is to extend the mentioned result to Frobenius algebras with semisimple Nakayama automorphism. The last section of the paper is devoted to present several examples of Frobenius algebras with semisimple Nakayama automorphism.
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