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Gerstenhaber Bracket on the Hochschild Cohomology via An Arbitrary Resolution

Published online by Cambridge University Press:  06 February 2019

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

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.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 2019 

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