Quasitriangular structures of the double of a finite group

TL;DR

This paper classifies all quasitriangular structures and ribbon elements of the Drinfeld double of a finite group, providing explicit descriptions and characterizations of braidings, equivalence classes, and factorizability.

Contribution

It offers an explicit classification of quasitriangular structures and ribbon elements of (G) in terms of group homomorphisms and central subgroups, advancing understanding of their braidings.

Findings

Explicit classification of quasitriangular structures and ribbon elements.

Characterization of equivalence classes under automorphisms.

Conditions for factorizability of structures.

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.