Let [Formula: see text] be a Hom–Hopf T-coalgebra over a group [Formula: see text] (i.e. a crossed Hom–Hopf [Formula: see text]-coalgebra). First, we introduce and study the left–right [Formula: see text]-Yetter–Drinfel’d category [Formula: see text] over [Formula: see text], with [Formula: see text], and construct a class of new braided T-categories. Then, we prove that a Yetter–Drinfel’d module category [Formula: see text] is a full subcategory of the center [Formula: see text] of the category of representations of [Formula: see text]. Next, we define the quasi-triangular structure of [Formula: see text] and show that the representation crossed category [Formula: see text] is quasi-braided. Finally, the Drinfel’d construction [Formula: see text] of [Formula: see text] is constructed, and an equivalent relation between [Formula: see text] and the representation of [Formula: see text] is given.

In this paper, we first give more interested properties of a weak Hopf quasigroup [Formula: see text]. Next, we introduce the notion of Yetter–Drinfel’d weak quasimodule over [Formula: see text] and prove that the category [Formula: see text] of right-right Yetter–Drinfel’d weak quasimodules is braided with left and right dualities under suitable conditions. Finally, we describe the notion of coquasitriangular weak Hopf quasigroup [Formula: see text], and study there exist a relation between Yetter–Drinfel’d weak quasimodule and the coinvariant space of right [Formula: see text]-comodule over [Formula: see text].

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Let H be a Hopf algebra and LR(H) the category of Yetter-Drinfel?d-Long bimodules over H. We first give sufficient and necessary conditions for LR(H) to be symmetry and pseudosymmetry, respectively. We then introduce the definition of the u-condition in LR(H) and discuss the relation between the u-condition and the symmetry of LR(H). Finally, we show that LR(H) over a triangular (cotriangular, resp.) Hopf algebra contains a rich symmetric subcategory.