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A Categorical Approach to Cyclic Cohomology of Quasi-Hopf Algebras and Hopf Algebroids

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Everything that’s extreme is difficult. The middle parts are done more easily. The very centre requires no effort at all. The centre is equal to equilibrium. There’s no fight in it.

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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.

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Acknowledgements

The authors wish to thank Masoud Khalkhali for stimulating questions and discussions. Furthermore, we are grateful to the referee for the constructive comments and useful references. The research of the second author was supported in part by the NSERC Discovery Grant Number 406709.

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Authors and Affiliations

  1. Department of Pure Mathematics, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

    Ivan Kobyzev

  2. Department of Mathematics and Statistics, University of Windsor, 401 Sunset Avenue, Windsor, ON, N9B 3P4, Canada

    Ilya Shapiro

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  1. Ivan Kobyzev
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  2. Ilya Shapiro
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Correspondence to Ivan Kobyzev.

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Communicated by M. Batanin.

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Kobyzev, I., Shapiro, I. A Categorical Approach to Cyclic Cohomology of Quasi-Hopf Algebras and Hopf Algebroids. Appl Categor Struct 27, 85–109 (2019). https://doi.org/10.1007/s10485-018-9544-0

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  • DOI: https://doi.org/10.1007/s10485-018-9544-0

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