Homology, Homotopy and Applications

Volume 18 (2016)

Number 1

An alternate approach to the Lie bracket on Hochschild cohomology

Pages: 265 – 285

DOI: https://dx.doi.org/10.4310/HHA.2016.v18.n1.a14


Cris Negron (Department of Mathematics, Louisiana State University, Baton Rouge, La., U.S.A.)

Sarah Witherspoon (Department of Mathematics, Texas A&M University, College Station, Tx., U.S.A.)


We define Gerstenhaber’s graded Lie bracket directly on complexes other than the bar complex, under some conditions, resulting in a practical technique for explicit computations. 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.


Hochschild cohomology, Gerstenhaber bracket, Koszul algebra

2010 Mathematics Subject Classification

16E40, 16S37, 18G10

Published 31 May 2016