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Gerstenhaber Bracket on the Hochschild Cohomology via An Arbitrary Resolution

Part of: Homological methods

Published online by Cambridge University Press:  06 February 2019

Yury Volkov
Yury Volkov*
Affiliation:
Saint-Petersburg State University, Universitetskaya nab. 7-9, St. Peterburg, Russia ([email protected])
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Abstract

We prove formulas of different types that allow us to calculate the Gerstenhaber bracket on the Hochschild cohomology of an algebra using some arbitrary projective bimodule resolution for it. Using one of these formulas, we give a new short proof of the derived invariance of the Gerstenhaber algebra structure on Hochschild cohomology. We also give some new formulas for the Connes differential on the Hochschild homology that lead to formulas for the Batalin–Vilkovisky (BV) differential on the Hochschild cohomology in the case of symmetric algebras. Finally, we use one of the obtained formulas to provide a full description of the BV structure and, correspondingly, the Gerstenhaber algebra structure on the Hochschild cohomology of a class of symmetric algebras.

Keywords

Hochschild cohomologyGerstenhaber bracketbimodule resolutionderived equivalence

MSC classification

Primary: 16E05: Syzygies, resolutions, complexes
Secondary: 16E35: Derived categories 16E40: (Co)homology of rings and algebras (e.g. Hochschild, cyclic, dihedral, etc.)
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 62 , Issue 3 , August 2019 , pp. 817 - 836
Copyright
Copyright © Edinburgh Mathematical Society 2019 

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