New braided \(T\)-categories over weak crossed Hopf group coalgebras (Q2318910)

Summary: Let \(H\) be a weak crossed Hopf group coalgebra over group \(\pi\); we first introduce a kind of new \(\alpha\)-Yetter-Drinfel'd module categories \(\mathcal{WYD}_{\alpha}(H)\) for \(\alpha \in \pi\) and use it to construct a braided \(T\)-category \(\mathcal{WYD}(H)\). As an application, we give the concept of a Long dimodule category \({}_H \mathcal{WL}^H\) for a weak crossed Hopf group coalgebra \(H\) with quasitriangular and coquasitriangular structures and obtain that \({}_H \mathcal{WL}^H\) is a braided \(T\)-category by translating it into a weak Yetter-Drinfel'd module subcategory \(\mathcal{WYD}(H \otimes H)\).