Constructing New Braided T-Categories over Weak Hopf Algebras

Let AutweakHopf(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 AutweakHopf(H)×AutweakHopf(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 ∈ AutweakHopf(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).