| The Hochschild cohomology HH•(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⊗A' of A with its Koszul dual A'.;We also investigate the Hochschild cohomology of smash product algebras A*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(A* 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* G. |
