Conmutatividad trenzada graduada en la cohomología de Hochschild.

The Hochschild cohomology of an algebra A over a field k has an associative product that is graded commutative, i.e., it is commutative up to signs depending on the degrees. This well-known result goes back to Gerstenhaber. If A is augmented, i.e., there is an algebra homomorphism from A to k, then k is a module via this homomorphism, termed the trivial module. The Hochschild cohomology of A with trivial coefficients also has an associative product, yet it need not be graded commutative. Versions of commutativity do hold under additional hypotheses. In this paper the authors extend what is known, thus capturing many more classes of examples. They were motivated by the known structure of cohomology of the Jordan plane and super Jordan plane. Placing these two algebras in a Yetter-Drinfeld category over the group algebra of the integers, the observed graded braided commutativity of their cohomology is now a consequence of the authors’ main theorems. These theorems establish graded braided commutativity, under some finiteness conditions, of Hochschild cohomology with trivial coefficients for a braided Hopf algebra A in the category of Yetter-Drinfeld modules over a Hopf algebra H. Specifically what this means is that the braiding on the category induces a braiding on relevant complexes that is applied, along with a sign, when switching the order of two cochains to obtain the same product in cohomology. The authors give several versions of this result, with different hypotheses on A and H: assume that A or H is finite dimensional as a vector space, or that H is an abelian group algebra and a projective resolution of A satisfies a certain finiteness condition. The authors’ results generalize those of Mastnak, Pevtsova, Schauenburg, and Witherspoon that relied heavily on existence of internal homs. Instead the authors here develop and use technology for complexes in duoidal categories. Their main results are for the duoidal category of bimodules over a Hopf algebra A in a braided monoidal category such as a Yetter-Drinfeld category. Under the various sets of finiteness conditions, they show graded braided commutativity of cohomology as a consequence of graded braided commutativity up to homotopy on complexes.