Quasitriangular structures of the double of a finite group

Abstract

We give a classification of all quasitriangular structures and ribbon elements of $\mathcal{D}(G)$ explicitly in terms of group homomorphisms and central subgroups. This can equivalently be interpreted as an explicit description of all braidings with which the tensor category $\operatorname{Rep}(\mathcal{D}(G))$ can be endowed. We also characterize their equivalence classes under the action of $\operatorname{Aut}(\mathcal{D}(G))$ and determine when they are factorizable.