Alternate Approaches to the Cup Product and Gerstenhaber Bracket on Hochschild Cohomology

Abstract

The Hochschild cohomology $HH^\bullet(A)$ of an algebra $A$ is a derived invariant of the algebra which admits both a graded ring structure (called the cup product) and a compatible graded Lie algebra structure (called the Gerstenhaber bracket). The Lie structure is particularly important as it provides a means of addressing the deformation theory of the algebra $A$. In this thesis we produce some new methods for analyzing the cup product and Gerstenhaber bracket on Hochschild cohomology. For the cup product we produce a number of new, and rather fundamental, relations between the theories of twisting cochains and Hochschild cohomology. In the case of a Koszul algebra $A$, our results imply that the Hochschild cohomology ring of $A$ is a subquotient of the tensor product algebra $A\ox A^!$ of $A$ with its Koszul dual $A^!$. We also investigate the Hochschild cohomology of smash product algebras $A\ast G$. (Here $A$ is an algebra equipped with an action of a Hopf algebra $G$.) In this setting, we produce new methods for computing both the cup product and Gerstenhaber bracket. For the Gerstenhaber bracket in particular, we show that there is an intermediate cohomology $H_{Int}^\bullet(A\ast G)$ which is a braided commutative algebra in the category of Yetter-Drinfeld modules over $G$, admits a braided anti-commutative bracket $[,]_{YD}$, and can be used to recover both the cup product and Gerstenhaber bracket on the standard Hochschild cohomology of $A\ast G$.