We prove the graded braided commutativity of the Hochschild cohomology of A with trivial coefficients, where A is a braided Hopf algebra in the category of Yetter-Drinfeld modules over the group algebra of an abelian group, under some finiteness conditions on a projective resolution of A as A-bimodule. This is a generalization of a result by Mastnak, Pevtsova, Schauenburg and Witherspoon to a context which includes Nichols algebras such as the Jordan and the super Jordan plane. We prove this result by constructing a coduoid-up-to-homotopy structure on the aforementioned projective resolution in the duoidal category of chain complexes of A-bimodules. We also prove that the Hochschild complex of a braided bialgebra A in an arbitrary braided monoidal category is a cocommutative comonoid up to homotopy with the deconcatenation product which induces the cup product in Hochschild cohomology.
Keywords: Hochschild cohomology, Nichols algebras, Hopf algebras, braided monoidal categories, duoidal categories
2020 MSC: 16B50, 16E40, 18G35, 18M15.
Theory and Applications of Categories, Vol. 41, 2024, No. 46, pp 1596-1643.
Published 2024-10-18.
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