Let $\pi $ be a group, and $H$ be a semi-Hopf $\pi $-algebra. We first show that the category $_H{\mathcal M}$ of left $\pi $-modules over $H$ is a monoidal category with a suitably defined tensor product and each element $\alpha $ in $\pi $ induces a strict monoidal functor $F_{\alpha }$ from $_H{\mathcal M}$ to itself. Then we introduce the concept of quasitriangular semi-Hopf $\pi $-algebra, and show that a semi-Hopf $\pi $-algebra $H$ is quasitriangular if and only if the category $_H\mathcal M$ is a braided monoidal category and $F_{\alpha }$ is a strict braided monoidal functor for any $\alpha \in \pi $. Finally, we give two examples of Hopf $\pi $-algebras and describe the categories of modules over them.
Let H be a finite-dimensional bialgebra. In this paper, we prove that the category LR(H) of Yetter-Drinfeld-Long bimodules, introduced by F.Panaite, F.Van Oystaeyen (2008), is isomorphic to the Yetter-Drinfeld category H⊗H H⊗H YD over the tensor product bialgebra H H∗ as monoidal categories. Moreover if H is a finite-dimensional Hopf algebra with bijective antipode, the isomorphism is braided. Finally, as an application of this category isomorphism, we give two results., Daowei Lu, Shuanhong Wang., and Seznam literatury