Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 9 (2013), 039, 36 pages      arXiv:1212.4879      http://dx.doi.org/10.3842/SIGMA.2013.039

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

Robert Coquereaux a, b and Jean-Bernard Zuber c
a) IMPA & UMI 2924 CNRS-IMPA, Jardim Botânico, Rio de Janeiro - RJ, 22460-320, Brazil
b) Aix Marseille Université, CNRS, CPT, UMR 7332, 13288 Marseille, France
c) LPTHE, CNRS-UMR 7589 and Université Pierre et Marie Curie, 4 place Jussieu, 75252, Paris Cedex 5, France

Received December 21, 2012, in final form May 15, 2013; Published online May 22, 2013; Misprints are corrected September 01, 2013

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], 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.

Key words: Lie group; fusion categories; conformal field theories; quantum symmetry; Drinfeld doubles.

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