Anti-Yetter-Drinfeld Modules for Quasi-Hopf Algebras

TL;DR

This paper develops a categorical framework to define anti-Yetter-Drinfeld modules for quasi-Hopf algebras, filling a gap in the theory of coefficients for Hopf cyclic cohomology in this context.

Contribution

It introduces the first definition of anti-Yetter-Drinfeld modules for quasi-Hopf algebras using categorical methods, extending the existing theory beyond Hopf algebras.

Findings

Provides a categorical construction for anti-Yetter-Drinfeld modules

Fills a gap in the theory of Hopf cyclic cohomology for quasi-Hopf algebras

Builds on previous work on anti-Yetter-Drinfeld contramodules

Abstract

We apply categorical machinery to the problem of defining anti-Yetter-Drinfeld modules for quasi-Hopf algebras. While a definition of Yetter-Drinfeld modules in this setting, extracted from their categorical interpretation as the center of the monoidal category of modules has been given, none was available for the anti-Yetter-Drinfeld modules that serve as coefficients for a Hopf cyclic type cohomology theory for quasi-Hopf algebras. This is a followup paper to the authors' previous effort that addressed the somewhat different case of anti-Yetter-Drinfeld contramodule coefficients in this and in the Hopf algebroid setting.