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

Abstract

Abstract. Analogous to a recent result of N. Kowalzig and U. Krähmer 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.

Batalin-Vilkovisky algebra structures on Hochschild cohomology of generalized Weyl algebras

“…Inspired by this thought, Kowalzig and Krähmer showed that if a skew Calabi-Yau algebra A has a semisimple Nakayama automorphism, then HH * (A) is a Batalin-Vilkovisky algebra, which is a generalization of Ginzburg's result [15]. Coincidentally, Lambre et al proved that for a Frobenius algebra A with semisimple Nakayama automorphism, HH * (A) is also a Batalin-Vilkovisky, generalizing Tradler's result [16]. In particular, when A is Koszul Calabi-Yau, the Koszul dual A !…”

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

“…Analogously, in 2016, Lambre, Zhou and Zimmermann proved in [15] that for a Frobenius algebra with semisimple Nakayama automorphism, say A ! , there also exists a Batalin-Vilkovisky algebra structure on HH • (A !…”

“…[15], Theorem 2.3). Let ∪ 1 , ∩ 1 and {−, −} 1 be the restrictions of the cup product, cap product and Gerstenhaber bracket to the homology and cohomology spaces associated with the eigenvalue λ = 1.…”

Maurer–Cartan elements and cyclic operads

“…If A is unital and comes equipped with an invariant symmetric nondegernate inner product, Connes' boundary operator induces a compatible BV operator [Men04], [Tra08]. More generally we may consider a (non-symmetric) Frobenius algebra with semi-simple Nakayama automorphism [LZZ14]. • In [BG10] Baranovsky and Ginzburg show that for a smooth complex Poisson variety X with smooth coisotropic subvarieties Y, Z, Tor O X (O Y , O Z ) is a Gerstenhaber algebra.…”