New Braided T‐Categories over Weak Crossed Hopf Group Coalgebras

Let H be a weak crossed Hopf group coalgebra over group π; we first introduce a kind of new α‐Yetter‐Drinfel’d module categories 𝒲𝒴𝒟α(H) for α ∈ π and use it to construct a braided T‐category 𝒲𝒴𝒟(H). As an application, we give the concept of a Long dimodule category H𝒲ℒH for a weak crossed Hopf group coalgebra H with quasitriangular and coquasitriangular structures and obtain that H𝒲ℒH is a braided T‐category by translating it into a weak Yetter‐Drinfel′d module subcategory 𝒲𝒴𝒟(H ⊗ H).