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

SIGMA 14 (2018), 098, 10 pages      arXiv:1804.02031

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

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.

pdf (305 kb)   tex (16 kb)


  1. Brzeziński T., Hopf-cyclic homology with contramodule coefficients, in Quantum Groups and Noncommutative Spaces, Aspects Math., Vol. E41, Vieweg + Teubner, Wiesbaden, 2011, 1-8, arXiv:0806.0389.
  2. Bulacu D., Panaite F., Van Oystaeyen F., Quantum traces and quantum dimensions for quasi-Hopf algebras, Comm. Algebra 27 (1999), 6103-6122.
  3. Connes A., Moscovici H., Hopf algebras, cyclic cohomology and the transverse index theorem, Comm. Math. Phys. 198 (1998), 199-246, math.DG/9806109.
  4. Connes A., Moscovici H., Cyclic cohomology and Hopf algebras, Lett. Math. Phys. 48 (1999), 97-108, math.QA/9904154.
  5. Drinfeld V.G., Quasi-Hopf algebras, Leningrad Math. J. 1 (1990), 1419-1457.
  6. Etingof P., Nikshych D., Ostrik V., Fusion categories and homotopy theory, Quantum Topol. 1 (2010), 209-273, arXiv:0909.3140.
  7. Hajac P.M., Khalkhali M., Rangipour B., Sommerhäuser Y., Hopf-cyclic homology and cohomology with coefficients, C. R. Math. Acad. Sci. Paris 338 (2004), 667-672, math.KT/0306288.
  8. Hajac P.M., Khalkhali M., Rangipour B., Sommerhäuser Y., Stable anti-Yetter-Drinfeld modules, C. R. Math. Acad. Sci. Paris 338 (2004), 587-590, math.QA/0405005.
  9. Hassanzadeh M., Khalkhali M., Shapiro I., Monoidal categories, 2-traces, and cyclic cohomology, Canad. Math. Bull., to appear, arXiv:1602.05441.
  10. Jara P., Ştefan D., Hopf-cyclic homology and relative cyclic homology of Hopf-Galois extensions, Proc. London Math. Soc. 93 (2006), 138-174, math.KT/0307099.
  11. Kobyzev I., Shapiro I., A categorical approach to cyclic cohomology of quasi-Hopf algebras and Hopf algebroids, arXiv:1803.09194.
  12. Majid S., Foundations of quantum group theory, Cambridge University Press, Cambridge, 1995.
  13. Majid S., Quantum double for quasi-Hopf algebras, Lett. Math. Phys. 45 (1998), 1-9, q-alg/9701002.
  14. Panaite F., Van Oystaeyen F., A structure theorem for quasi-Hopf comodule algebras, Proc. Amer. Math. Soc. 135 (2007), 1669-1677, math.QA/0506272.
  15. Sakáloš Š., On categories associated to a quasi-Hopf algebra, Comm. Algebra 45 (2017), 722-748, arXiv:1402.1393.
  16. Shapiro I., Some invariance properties of cyclic cohomology with coefficients, arXiv:1611.01425.

Previous article  Next article   Contents of Volume 14 (2018)