Graded braided commutativity in Hochschild cohomology

TL;DR

This paper proves graded braided commutativity of Hochschild cohomology for certain braided Hopf algebras, extending previous results to include Nichols algebras and constructing a homotopy structure on projective resolutions.

Contribution

It generalizes the graded braided commutativity result to a broader class of braided Hopf algebras, including Nichols algebras, using a homotopy-based approach.

Findings

Hochschild cohomology is graded braided commutative under specified conditions.

Constructed a coduoid-up-to-homotopy structure on projective resolutions.

Hochschild complex of a braided bialgebra is a cocommutative comonoid up to homotopy.

Abstract

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.