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
@article{TAC_2024_41_a45,
author = {Javier C\'oppola and Andrea Solotar},
title = {Graded braided commutativity in {Hochschild} cohomology},
journal = {Theory and applications of categories},
pages = {1596--1643},
year = {2024},
volume = {41},
language = {en},
url = {http://geodesic.mathdoc.fr/item/TAC_2024_41_a45/}
}
Javier Cóppola; Andrea Solotar. Graded braided commutativity in Hochschild cohomology. Theory and applications of categories, Tome 41 (2024), pp. 1596-1643. http://geodesic.mathdoc.fr/item/TAC_2024_41_a45/