Abstract
For a quasi-Hopf algebra H, an H-bicomodule algebra \(\mathbb{A}\) and an H-bimodule coalgebra C we will show that the category of two-sided two-cosided Hopf modules \({}^C_H{\cal M}_{\mathbb{A}}^H\) is equivalent to the category of right–left generalized Yetter–Drinfeld modules \({}^C{\cal YD}(H)_{\mathbb{A}}\). Using alternative versions of this result we will recover the category isomorphism between the categories of left–left and left–right Yetter–Drinfeld modules over a quasi-Hopf algebra.
Similar content being viewed by others
References
Beattie, M., Dăscălescu, Ş., Raianu, Van Oystaeyen, F.: The categories of Yetter–Drinfeld modules, Doi–Hopf modules and two-sided two-cosided Hopf modules. Appl. Categor. Struct. 6, 223–237 (1998)
Bulacu, D., Caenepeel, S.: Two-sided two-cosided Hopf modules and Doi–Hopf modules for quasi-Hopf algebras. J. Algebra 270(1), 55–95 (2003)
Bulacu, D., Caenepeel, S.: Integrals for (dual) quasi-Hopf algebras. J. Algebra 266(2), 552–583 (2003)
Bulacu, D., Nauwelaerts, E.: Relative Hopf modules for (dual)quasi Hopf algebras. J. Algebra 229, 632–659 (2000)
Bulacu, D., Panaite, F., Van Oystaeyen, F.: Quasi-Hopf algebra actions and smash products. Comm. Algebra 28, 631–651 (2000)
Bulacu, D., Caenepeel, S., Panaite, F.: More properties of Yetter–Drinfeld modules over quasi-Hopf algebras. In: Caenepeel, S., Van Oystaeyen, F. (eds.) Hopf algebras in non-commutative geometry and physics. Lecture Notes Pure and Applied Mathematics, vol. 239, pp. 89–112, Dekker, New York (2005)
Bulacu, D., Caenepeel, S., Panaite, F.: Yetter–Drinfeld categories for quasi-Hopf algebras. Comm. Algebra 34(1), 1–35 (2006)
Bulacu, D., Caenepeel, S., Torrecillas, B.: Doi–Hopf modules and Yetter–Drinfeld modules for quasi-Hopf algebras. Comm. Algebra 34, 3413–3449 (2006)
Bulacu, D., Panaite, F., Van Oystaeyen, F.: Generalized diagonal crossed products and smash products for (quasi) Hopf algebras. Comm. Math. Phys. 266, 355–399 (2006)
Drinfeld, V.G.: Quasi-Hopf algebras. Leningr. Math. J. 1, 1419–1457 (1990)
Hausser, F., Nill, F.: Diagonal crossed products by duals of quasi-quantum groups. Rev. Math. Phys. 11, 553–629 (1999)
Hausser F., Nill, F.: Integral theory for quasi-Hopf algebras, preprint math. QA/9904164.
Kassel, C.: Quantum groups. In: Graduate Texts in Mathematics, vol. 155. Springer, Berlin Heidelberg New York (1995)
Majid, S.: Quantum double for quasi-Hopf algebras. Lett. Math. Phys. 45, 1–9 (1998)
Majid, S.: Foundations of Quantum Group Theory. Cambridge University Press, UK (1995)
Schauenburg, P.: Hopf modules and the double of a quasi-Hopf algebra. Trans. Amer. Math. Soc. 354(8), 3349–3378 (2002)
Schauenburg, P.: Hopf modules and Yetter–Drinfeld modules. J. Algebra 169, 874–890 (1994)
Sweedler, M.E.: Hopf Algebras. Benjamin, New York (1969)
Tambara, D.: The bialgebra of coendomorphism of an algebra. J. Fac. Sci., Univ. Tokyo, Sect. 1A Math. 37, 425–456 (1990)
Woronowicz, S.L.: Differential calculus on compact matrix pseudogroups (quantum groups). Comm. Math. Phys. 122, 125–170 (1989)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bulacu, D., Torrecillas, B. Two-sided Two-cosided Hopf Modules and Yetter–Drinfeld Modules for Quasi-Hopf Algebras. Appl Categor Struct 14, 503–530 (2006). https://doi.org/10.1007/s10485-006-9045-4
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10485-006-9045-4