New Braided T-Categories over Weak Crossed Hopf Group Coalgebras

Abstract

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