Quasitriangular Hopf group algebras and braided monoidal categories

Abstract

Let π be a group, and H be a semi-Hopf π-algebra. We first show that the category H M of left π-modules over H is a monoidal category with a suitably defined tensor product and each element α in π induces a strict monoidal functor $${F_\alpha }$$ from H M to itself. Then we introduce the concept of quasitriangular semi-Hopf π-algebra, and show that a semi-Hopf π-algebra H is quasitriangular if and only if the category H Mis a braided monoidal category and $${F_\alpha }$$ is a strict braided monoidal functor for any α ∈ π. Finally, we give two examples of Hopf π-algebras and describe the categories of modules over them.