Batalin–Vilkovisky structures on Ext and Tor

Abstract

This article studies the algebraic structure of homology theories defined by a left Hopf algebroid U over a possibly noncommutative base algebra A, such as for example Hochschild, Lie algebroid (in particular Lie algebra and Poisson), or group and etale groupoid (co)homology. Explicit formulae for the canonical Gerstenhaber algebra structure on Ext U pA, Aq are given. The main technical result constructs a Lie derivative satisfying a generalised Cartan-Rinehart homotopy formula whose essence is that Tor U pM, Aq becomes for suitable right U -modules M a Batalin-Vilkovisky module over Ext U pA, Aq, or in the words of Nest, Tamarkin, Tsygan and others, that Ext U pA, Aq and Tor U pM, Aq form a differential calculus. As an illustration, we show how the wellknown operators from differential geometry in the classical Cartan homotopy formula can be obtained. Another application consists in generalising Ginzburg's result that the cohomology ring of a Calabi-Yau algebra is a Batalin-Vilkovisky algebra to twisted Calabi-Yau algebras.

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

“…In this section, we briefly recall the construction of the Batalin-Vilkovisky algebra on the Hochschild cohomology of Artin-Schelter regular algebras with semisimple Nakayama automorphism, obtained by Kowalzig and Krahmer in [13]. In this section, A will present a connected graded algebra over an algebraically closed field k. A graded algebra A is said to be connected if A i = 0 for i < 0 and A 0 = k.…”

“…Assume A is an AS-regular algebra with semisimple Nakayama automorphism σ. In [12,13], Kowalzig and Krahmer constructed a differential calculus with duality on (HH • (A), HH • (A; A σ )). Thus as a corollary, they obtained a Batalin-Vilkovisky algebra structure on HH • (A).…”

“…In 2014, Kowalzig and Krahmer showed in [13] that the Hochschild cohomology of an Artin-Schelter (AS for short) regular algebra with semisimple Nakayama automorphism has a Batalin-Vilkovisky algebra structure. Soon after that, Lambre, Zhou and Zimmerman proved in [15] that the Hochschild cohomology of a Frobenius algebra with semisimple Nakayama automorphism also admits a Batalin-Vilkovisky structure.…”

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

“…Kowalzig and U. Krähmer extended the result of V. Ginzburg to twisted Calabi-Yau algebras under a certain condition Let A be a twisted Calabi-Yau algebra with semisimple algebra automorphism σ. Then the Hochschild cohomology ring of A is a Batalin-Vilkovisky algebra; see [24,Theorem 1.7].…”

“…This notion explains when BV structure exists and unifies the two known cases of symmetric algebras and Calabi-Yau algebras. Recently as an application of this notion, N. Kowalzig and U. Krähmer [24,Theorem 1.7] proved that the Hochschild cohomology ring of a twisted Calabi-Yau algebra is also a Batalin-Vilkovisky algebra, provided a certain algebra automorphism is semisimple.…”