The Hochschild cohomology ring of a Frobenius algebra with semisimple Nakayama automorphism is a Batalin-Vilkovisky algebra

TL;DR

This paper proves that the Hochschild cohomology ring of certain Frobenius algebras possesses a Batalin-Vilkovisky algebra structure, extending previous results and providing criteria and examples for such algebras.

Contribution

It generalizes the Batalin-Vilkovisky structure result to Frobenius algebras with semisimple Nakayama automorphism and offers criteria and examples for these algebras.

Findings

Hochschild cohomology ring is a BV algebra under specified conditions

Provides criteria for semisimple Nakayama automorphism in quiver algebras

Includes examples like quantum complete intersections and Hopf algebras

Abstract

Analogous to a recent result of N. Kowalzig and U. Kr\"{a}hmer for twisted Calabi-Yau algebras, we show that the Hochschild cohomology ring of a Frobenius algebra with semisimple Nakayama automorphism is a Batalin-Vilkovisky algebra, thus generalizing a result of T.Tradler for finite dimensional symmetric algebras. We give a criterion to determine when a Frobenius algebra given by quiver with relations has semisimple Nakayama automorphism and apply it to some known classes of tame Frobenius algebras. We also provide ample examples including quantum complete intersections, finite dimensional Hopf algebras defined over an algebraically closed field of characteristic zero and Koszul duals of Koszul Artin-Schelter regular algebras of dimension three.