A categorical approach to cyclic cohomology of quasi-Hopf algebras and Hopf algebroids

TL;DR

This paper develops a categorical framework for defining cyclic cohomology with coefficients for quasi-Hopf algebras and Hopf algebroids, providing new insights and clarifications for these algebraic structures.

Contribution

It introduces a novel categorical approach to cyclic cohomology for quasi-Hopf algebras, and clarifies existing formulas for Hopf algebroids using biclosed monoidal categories.

Findings

Defined cyclic cohomology for quasi-Hopf algebras for the first time.

Provided a clearer, more natural formulation for Hopf algebroids.

Highlighted the role of biclosed monoidal categories in cyclic theory.

Abstract

We apply categorical machinery to the problem of defining cyclic cohomology with coefficients in two particular cases, namely quasi-Hopf algebras and Hopf algebroids. In the case of the former, no definition was thus far available in the literature, and while a definition exists for the latter, we feel that our approach demystifies the seemingly arbitrary formulas present there. This paper emphasizes the importance of working with a biclosed monoidal category in order to obtain natural coefficients for a cyclic theory that are analogous to the stable anti-Yetter-Drinfeld contramodules for Hopf algebras.