Drinfeld Doubles for Finite Subgroups of SU(2) and SU(3) Lie Groups

TL;DR

This paper computes the modular data of Drinfeld doubles of finite subgroups of SU(2) and SU(3), revealing that certain fusion identities do not hold, and provides extensive data relevant to quantum algebra and conformal field theory.

Contribution

It provides explicit calculations of modular data for Drinfeld doubles of specific finite groups and challenges assumptions about fusion identity validity.

Findings

Fusion identities fail for Drinfeld doubles, similar to finite groups.

Explicit modular data for these doubles are computed and tabulated.

Results have implications for orbifold theories in conformal field theory.

Abstract

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 our recent paper [J. Phys. A: Math. Theor. 44 (2011), 295208, 26 pages, 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 which are interesting for their own sake and maybe also in a relation with the theory of orbifolds in conformal field theory.