### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 098, 10 pages      arXiv:1804.02031      https://doi.org/10.3842/SIGMA.2018.098

### Anti-Yetter-Drinfeld Modules for Quasi-Hopf Algebras

Ivan Kobyzev a and Ilya Shapiro b
a) Department of Pure Mathematics, University of Waterloo, Waterloo, ON N2L 3G1, Canada
b) Department of Mathematics and Statistics, University of Windsor, 401 Sunset Avenue, Windsor, Ontario N9B 3P4, Canada

Received April 20, 2018, in final form September 10, 2018; Published online September 13, 2018

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.

Key words: monoidal category; cyclic homology; Hopf algebras; quasi-Hopf algebras.

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