Abstract
Let Aut weak Hopf (H) denote the set of all automorphisms of a weak Hopf algebra H with bijective antipode in the sense of Böhm et al. (J Algebra 221:385–438, 1999) and let G be a certain crossed product group Aut weak Hopf (H)×Aut weak Hopf (H). The main purpose of this paper is to provide further examples of braided T-categories in the sense of Turaev (1994, 2008). For this, we first introduce a class of new categories \( _{H}{\mathcal {WYD}}^{H}(\alpha, \beta)\) of weak (α, β)-Yetter-Drinfeld modules with α, β ∈ Aut weak Hopf (H) and we show that the category \({\mathcal WYD}(H) =\{{}_{H}\mathcal {WYD}^{H}(\alpha, \beta)\}_{(\alpha , \beta )\in G}\) becomes a braided T-category over G, generalizing the main constructions by Panaite and Staic (Isr J Math 158:349–365, 2007). Finally, when H is finite-dimensional we construct a quasitriangular weak T-coalgebra WD(H) = {WD(H)(α, β)}(α, β) ∈ G in the sense of Van Daele and Wang (Comm Algebra, 2008) over a family of weak smash product algebras \(\{\overline{H^{*cop}\# H_{(\alpha,\beta)}}\}_{(\alpha , \beta)\in G}\), and we obtain that \({\mathcal {WYD}}(H)\) is isomorphic to the representation category of the quasitriangular weak T-coalgebra WD(H).
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
Böhm, G., Nill, F., Szlachányi, K.: Weak Hopf algebras I: integral theory and C*-structure. J. Algebra 221, 385–438 (1999)
Böhm, G.: Doi-Hopf modules over weak Hopf algebras. Comm. Algebra 28, 4687–4698 (2000)
Caenepeel, S., De Lombaerde, M.: A categorical approach to Turaev’s Hopf group-coalgebras. Comm. Algebra 34, 2631–2657 (2006)
Caenepeel, S., De Groot, E.: Modules over weak entwining structures. Contemp. Math. 267, 31–54 (2000)
Caenepeel, S., Van Oystaeyen, F., Zhang, Y.H.: The Brauer group of Yetter-Drinfeld module algebras. Trans. Am. Math. Soc. 349, 3737–3771 (1997)
Caenepeel, S., Wang, D.G., Yin, Y.M.: Yetter-Drinfeld modules over weak bialgebras. Ann. Univ. Ferrara-Sez. VII-Sc. Mat. 51, 69–98 (2005)
Freyd, P.J., Yetter, D.N.: Braided compact closed categories with applications to low-dimensional topology. Adv. Math. 77, 156–182 (1989)
Hajac, P.M., Khalkhali, M., Rangipour, B., Sommerhauser, Y.: Stable anti-Yetter-Drinfeld modules. C. R. Math. Acad. Sci. Paris 338, 587–590 (2004)
Hajac, P.M., Khalkhali, M., Rangipour, B., Sommerhauser, Y.: Hopf-cyclic homology and cohomology with coefficients. C. R. Math. Acad. Sci. Paris 338, 667–672 (2004)
Jara, P., Stefan, D.: Hopf-cyclic homology and relative cyclic homology of Hopf-Galois extensions. Proc. Lond. Math. Soc. 93, 138–174 (2006)
Joyal, A., Street, R.: Braided tensor categories. Adv. Math. 102, 20–78 (1993)
Nikshych, D., Turaev, V., Vainerman, L.: Invariants of knots and 3-manifolds from quantum groupoids. Topol. its Appl. 127, 91–123 (2003)
Nikshych, D., Vainerman, L.: Finite quantum groupoids and their applications. In: Montgomery, S., Schneider, H.-J. (eds.), New Directions in Hopf Algebras. Math. Sci. Res. Inst. Publ. 43, pp. 211–262. Cambridge University Press, Cambridge (2002)
Panaite, F., Staic, M.D.: Generalized (anti) Yetter-Drinfeld modules as components of a braided T-category. Isr. J. Math. 158, 349–365 (2007)
Sweedler, M.: Hopf Algebras. Benjamin, New York (1969)
Turaev, V.G.: Quantum Invariants of Knots and 3-Manifolds. de Gruyter Stud. Math. Vol. 18. de Gruyter, Berlin (1994)
Turaev, V.G.: Homotopy field theory in dimension 3 and crossed group-categories. GT/0005291 (2008)
Van Daele, A., Wang, S.H.: New braided crossed categories and Drinfeld quantum double for weak Hopf π-coalgebras. Comm. Algebra 36, 2341–2386 (2008)
Virelizier, A.: Hopf group-coalgebras. J. Pure Appl. Algebra 171, 75–122 (2002)
Zunino, M.: Yetter-Drinfeld modules for crossed structures. J. Pure Appl. Algebra 193, 313–343 (2004)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Liu, L., Wang, S. Constructing New Braided T-Categories over Weak Hopf Algebras. Appl Categor Struct 18, 431–459 (2010). https://doi.org/10.1007/s10485-008-9175-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10485-008-9175-y
Keywords
- Weak Hopf algebra
- Braided T-category
- Weak (α, β)-Yetter-Drinfeld modules
- Quasitriangular weak T-coalgebra