Advertisement

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
Article
  • 29 Downloads

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.

Keywords

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

Mathematics Subject Classification

18D10 18E05 19D55 16T05 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

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.

References

  1. 1.
    Böhm, G.: Hopf Algebroids, Handbook of Algebra Vol 6, edited by M. Hazewinkel, Elsevier, (2009), pp. 173–236Google Scholar
  2. 2.
    Böhm, G., Stefan, D.: A categorical approach to cyclic duality. J. Noncommutative Geom. 6(3), 481–538 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Brzezinski, T.: Hopf-cyclic homology with contramodule coefficients, Quantum groups and noncommutative spaces, 1–8, Aspects Math., E41, Vieweg + Teubner, Wiesbaden, (2011)Google Scholar
  4. 4.
    Bulacu, D., Panaite, F., Van Oystaeyen, F.: Quantum traces and quantum dimensions for quasi-Hopf algebras. Commun. Algebra 27(12), 6103–6122 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Connes, A., Moscovici, H.: Hopf algebras, cyclic cohomology and the transverse index theorem. Commun. Math. Phys. 198, 199–246 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Connes, A., Moscovici, H.: Cyclic cohomology and Hopf algebras. Lett. Math. Phys. 48, 97–108 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Drinfeld, V.: Quasi-Hopf algebras. Leningr. Math. J. 1(6), 1419–1457 (1990)MathSciNetGoogle Scholar
  8. 8.
    Eilenberg, S., Moore, J.C.: Foundations of Relative Homological Algebra, vol. 55. American Mathematical Society, Providence (1965)zbMATHGoogle Scholar
  9. 9.
    Gelaki, S., Naidu, D., Nikshych, D.: Centers of graded fusion categories. Algebra Number Theory 3(8), 959–990 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hajac, P.M., Khalkhali, M., Rangipour, B., Sommerhauser, Y.: Hopf-cyclic homology and cohomology with coefficients. C. R. Math. Acad. Sci. Paris 338(9), 667–672 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hajac, P.M., Khalkhali, M., Rangipour, B., Sommerhäuser, Y.: Stable anti-Yetter–Drinfeld modules. C. R. Acad. Sci. Paris, Ser. I 338, 587–590 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hassanzadeh, M., Khalkhali, M., Shapiro, I.: Monoidal categories, 2-traces, and cyclic cohomology. arXiv:1602.05441
  13. 13.
    Jara, P., Stefan, D.: Hopf–cyclic homology and relative cyclic homology of Hopf–Galois extensions. Proc. London Math. Soc. 93(1), 138–174 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kobyzev, I., Shapiro, I.: Anti-Yetter–Drinfeld modules for quasi-Hopf algebras. arXiv:1804.02031
  15. 15.
    Kowalzig, N.: When Ext is a Batalin–Vilkovisky algebra (2016). arXiv:1610.01229
  16. 16.
    Kowalzig, N., Posthuma, H.: The cyclic theory of Hopf algebroids. J. Noncommutative Geometry 5(3), 423–476 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Majid, S.: Foundations of Quantum Group Theory. Cambridge University Press, Cambridge (1995)CrossRefzbMATHGoogle Scholar
  18. 18.
    Majid, S.: Quantum double for Quasi–Hopf algebras. Lett Math Phys 45(1), 1–9 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Sakáloš, Š.: On categories associated to a Quasi-Hopf algebra. Commun. Algebra 45(2), 722–748 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Schauenburg, P.: Bialgebras over noncommutative rings and a structure theorem for Hopf bimodules. Appl. Categ. Struct. 6, 193–222 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Schauenburg, P.: Duals and doubles of quantum groupoids (\(\times _R\)-Hopf algebras). AMS Contemp. Math. 267, 273–299 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Shapiro, I.: Some invariance properties of cyclic cohomology with coefficients. arXiv:1611.01425

Copyright information

© 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

Personalised recommendations