Applied Categorical Structures

, Volume 27, Issue 1, pp 85–109 | Cite as

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

  • Ivan KobyzevEmail author
  • Ilya Shapiro


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.


Cyclic homology Hopf algebras Quasi-Hopf algebras Hopf algebroids Monoidal categories Contramodules 

Mathematics Subject Classification

18D10 18E05 19D55 16T05 


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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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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of Pure MathematicsUniversity of WaterlooWaterlooCanada
  2. 2.Department of Mathematics and StatisticsUniversity of WindsorWindsorCanada

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