Brackets on Hochschild cohomology of Noncommutative Algebras

Abstract

The Hochschild cohomology of an associative algebra is a Gerstenhaber algebra, having a graded ring structure given by the cup product and a compatible graded Lie algebra structure given by the Gerstenhaber bracket. The cup product can be defined generally from multiple perspectives and has been studied for many classes of algebras. The Gerstenhaber bracket, however, has not admitted such a general definition, making computations difficult.

In this dissertation, we characterize the Gerstenhaber algebra structure on the Hochschild cohomology of group extensions of quantum complete intersections. We utilize the notion of twisted tensor products, a noncommutative tensor product, and adapt a technique of Wambst’s to compute the graded ring structure on Hochschild cohomology. The bracket structure is computed by employing an alternative description given in recent work of Negron and Witherspoon. When the group is trivial, this work extends the previous computations of the graded ring structure of Hochschild cohomology of quantum complete intersections to include the bracket structure. As an example, we compute the Gerstenhaber algebra structure for two generator quantum complete intersections extended by selected groups.