Gerstenhaber bracket on the Hochschild cohomology via an arbitrary resolution

TL;DR

This paper develops formulas to compute the Gerstenhaber bracket on Hochschild cohomology using arbitrary resolutions, providing new proofs and descriptions of algebraic structures for symmetric algebras.

Contribution

It introduces new formulas for calculating the Gerstenhaber bracket and BV differential using arbitrary resolutions, and applies these to symmetric algebras.

Findings

Formulas for Gerstenhaber bracket via arbitrary resolutions

A new proof of derived invariance of Hochschild cohomology structure

Full description of BV and Gerstenhaber structures for certain symmetric algebras

Abstract

We prove formulas of different types that allow 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. Also we give some new formulas for the Connes' differential on the Hochschild homology that lead to formulas for BV differential on the Hochschild cohomology in the case of symmetric algebras. Finally, we use one of the obtained formulas to get a full description of the BV structure and, correspondingly, the Gerstenhaber algebra structure on the Hochschild cohomology of a class of symmetric algebras.