The Hochschild cohomology ring of a Frobenius algebra with semisimple Nakayama automorphism is a Batalin-Vilkovisky algebra.
type: Item
instance of: scholarly article
title: The Hochschild cohomology ring of a Frobenius algebra with semisimple Nakayama automorphism is a Batalin-Vilkovisky algebra.
zbMATH Open document ID: 1344.16009
author: Thierry Lambre
author: Guodong Zhou
author: Alexander Zimmermann
published in: Journal of Algebra
publication date: 2015-11-17T00:00:00Z
full work available at URL: https://arxiv.org/abs/1405.5325
review text: 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.
Mathematics Subject Classification ID: 16E40
Mathematics Subject Classification ID: 16L60
Mathematics Subject Classification ID: 16S38
zbMATH DE Number: 6509812
zbMATH Keywords: symmetric algebras
zbMATH Keywords: Calabi-Yau algebras
zbMATH Keywords: Batalin-Vilkovisky algebras
zbMATH Keywords: differential calculi with duality
zbMATH Keywords: Frobenius algebras
zbMATH Keywords: Hochschild cohomology
zbMATH Keywords: semisimple Nakayama automorphisms
zbMATH Keywords: Gerstenhaber structures
Wikidata QID: Q112881620
reviewed by: Blas Torrecillas
MaRDI profile type: Publication
OpenAlex ID: W2963225378
arXiv ID: 1405.5325
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DOI: 10.1016/J.JALGEBRA.2015.09.018
description: scientific article