The gerstenhaber bracket in hochschild cohomology: methods and examples

Abstract

Homological methods provide important information about the structureof associative algebras, revealing sometimes hidden connections amongst them. Thistext is about the Gerstenhaber bracket in Hochschild cohomology of unital associativealgebras over a eld. The rst Hochschild cohomology space of an associative algebra isa Lie algebra with the Gerstenhaber bracket. The computation of Hochschild cohomologyrequires a resolution of the algebra considered as a bimodule over itself. The barresolution is not very satisfactory for explicit calculations to be carried out. The useof alternative resolutions is not well adapted to the computation of the Gerstenhaberbracket. However, some results by Witherspoon-Negron, Volkov and Suarez Alvarezprovide useful tools to solve this problem. I will illustrate how, using these methods,it is possible to describe the rst Hochschild cohomology spaces of some families ofalgebras as Lie algebras, the Lie module structure of higher Hochschild cohomologyspaces, and relate this to the structure of the algebras.