Symmetries in Yetter-Drinfel’d-Long categories

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.