Drinfeld doubles for finite subgroups of SU(2) and SU(3) Lie groups
Résumé
Drinfeld doubles of finite subgroups of SU(2) and SU(3) are investigated in detail. Their modular data --S, T and fusion matrices-- are computed explicitly, and illustrated by means of fusion graphs. This allows us to reexamine certain identities on these tensor product or fusion multiplicities under conjugation of representations that had been discussed in a recent paper [Coquereaux-Zuber 2011 arXiv:1103.2943], proved to hold for simple and affine Lie algebras, and found to be generally wrong for finite groups. It is shown here that these identities fail also in general for Drinfeld doubles, indicating that modularity of the fusion category is not the decisive feature. Along the way, we collect many data on these Drinfeld doubles and make a certain number of curious observations.