Quasi Triangle Hopf Pi - Algebra

Quantum groups were defined by Drinfeld’s terminology as the quasitriangular Hopf algebras. The category of modules over a quasitriangular Hopf algebra is a braided monoidal category. Moreover, the braiding structure of a braided monoidal category can supply solutions to the quantum Yang-Baxter equation. Dually, the category of comodules over a coquasitriangular Hopf algebra is also a braided monoidal category. These concepts and theories have a strong physical background, and have close relations with other branches of mathematics. The theory has achieved great development so far. During the past several years, many mathematicians have been working on Hopf algebras and braided monoidal categories. Turaev introduced the concepts of π-coalgebra, Hopf π-coalgebra and the quasitriangular Hopf π-coalgebra in [16], where π is a group. Virelizier studied some basic properties of Hopf π-coalgebras and introduced the concept of π-comodule and generalized the main propoerties of the quasitriangular Hopf algebras to quasitriangular Hopf π-coalgebras in [17].Based on the above background, we studied the categories of modules over Hopf π-algebras and their endofunctors. We first investigate module category over a Hopf π-algebra H. We introduce the concepts of the left H-π-modules and the H-π-module homomorphisms. We show that the category (H-Mod,(?), K, a,1, r) of the left H-π-modules is a monoidal category and each element α in π induces a monoidal functor (Fa,id,id) from (H-Mod,(?), K, a, l, r) to itself. Then we introduce the concept of quasitriangular Hopf π-algebras and show that a Hopf π-algebra H is a quasitriangular Hopf π-algebra if and only if the category (H-Mod,(?), K, a,l, r) of the left H-π-modules is a braided monoidal category and(Fα,id,id) is a braided monoidal functor for all α∈π.