An Alternate Approach to the Lie Bracket on Hochschild Cohomology

TL;DR

This paper introduces a new method to define the Lie bracket on Hochschild cohomology directly on complexes other than the bar complex, broadening the scope of applications and simplifying computations.

Contribution

It presents a novel approach to defining the Gerstenhaber Lie bracket on complexes like the Koszul complex, extending known brackets such as Schouten-Nijenhuis and Gerstenhaber brackets.

Findings

Successfully defines the Lie bracket on Koszul complexes

Recovers classical brackets for polynomial rings and cyclic group algebras

Broadens the applicability of Hochschild cohomology computations

Abstract

We define Gerstenhaber's graded Lie bracket directly on complexes other than the bar complex, under some conditions. The Koszul complex of a Koszul algebra in particular satisfies our conditions. As examples we recover the Schouten-Nijenhuis bracket for a polynomial ring and the Gerstenhaber bracket for a group algebra of a cyclic group of prime order.