Koszul duality and the Hochschild cohomology of Artin–Schelter regular algebras

Abstract

We identify two Batalin-Vilkovisky algebra structures, one obtained by Kowalzig and Krahmer on the Hochschild cohomology of an Artin-Schelter regular algebra with semisimple Nakayama automorphism and the other obtained by Lambre, Zhou and Zimmermann on the Hochschild cohomology of a Frobenius algebra also with semisimple Nakayama automorphism, provided that these two algebras are Koszul dual to each other.

Batalin-Vilkovisky algebra structure on Poisson manifolds with semi-simple modular symmetry

“…In this paper we show that the above results remain true if the modular vector of the Poisson algebra is semisimple. We remark that the Batalin-Vilkovisky algebra structures for AS-regular algebras and for Frobenius algebras with semisimple Nakayama were independently proved by Kowalzig and Krahmer [22] and Lambre, Zhou and Zimmermann [25] respectively; their isomorphism in the Koszul case was proved by [29]. What is new here is that we relate these results with those structures appearing in deformation quantization, and hence gives a direct connection with Poisson geometry, and in particular, with the two results obtained by Luo, Wang and Wu in [30,31], where Poisson algebras with nontrivial modular symmetry were studied.…”

“…Proof. The isomorphism of Batalin-Vilkovisky algebras on the Hochschild cohomology is proved in [29,Theorem 1.1]. The isomorphism of gravity algebras on the cyclic homology then follows from combining the above isomorphism with Lemma 4.2.…”