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

SIGMA 9 (2013), 039, 36 pages      arXiv:1212.4879

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

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.

pdf (1472 kb)   tex (923 kb)       [previous version:  pdf (1462 kb)   tex (923 kb)]


  1. Behrend R.E., Pearce P.A., Petkova V.B., Zuber J.-B., Boundary conditions in rational conformal field theories, Nuclear Phys. B 579 (2000), 707-773, hep-th/9908036.
  2. Blichfeldt H.F., Finite collineation groups, The University Chicago Press, Chicago, 1917.
  3. Cappelli A., D'Appollonio G., Boundary states of c=1 and c=3/2 rational conformal field theories, J. High Energy Phys. 2002 (2002), no. 2, 039, 45 pages, hep-th/0201173.
  4. Cardy J.L., Boundary conditions, fusion rules and the Verlinde formula, Nuclear Phys. B 324 (1989), 581-596.
  5. Cibils C., Tensor product of Hopf bimodules over a group, Proc. Amer. Math. Soc. 125 (1997), 1315-1321.
  6. Coquereaux R., Character tables (modular data) for Drinfeld doubles of finite groups, PoS Proc. Sci. (2012), PoS(ICMP2012), 024, 8 pages, arXiv:1212.4010.
  7. Coquereaux R., Isasi E., Schieber G., Notes on TQFT wire models and coherence equations for SU(3) triangular cells, SIGMA 6 (2010), 099, 44 pages, arXiv:1007.0721.
  8. Coquereaux R., Zuber J.-B., On sums of tensor and fusion multiplicities, J. Phys. A: Math. Theor. 44 (2011), 295208, 26 pages, arXiv:1103.2943.
  9. Coste A., Gannon T., Ruelle P., Finite group modular data, Nuclear Phys. B 581 (2000), 679-717, hep-th/0001158.
  10. Covering groups of the alternating and symmetric groups,
  11. Curtis C.W., Reiner I., Representation theory of finite groups and associative algebras, Pure and Applied Mathematics, Vol. 11, Interscience Publishers, New York - London, 1962.
  12. Di Francesco P., Zuber J.-B., SU(N) lattice integrable models associated with graphs, Nuclear Phys. B 338 (1990), 602-646.
  13. Dijkgraaf R., Pasquier V., Roche P., Quasi Hopf algebras, group cohomology and orbifold models, Nuclear Phys. B Proc. Suppl. 18 (1991), 60-72.
  14. Dijkgraaf R., Vafa C., Verlinde E., Verlinde H., The operator algebra of orbifold models, Comm. Math. Phys. 123 (1989), 485-526.
  15. Drinfeld V.G., Quasi-Hopf algebras, Leningrad Math. J. 1 (1989), 1419-1457.
  16. Drinfeld V.G., Quasi-Hopf algebras and Knizhnik-Zamolodchikov equations, Preprint ITP-89-43E, Kiev, 1989.
  17. Drinfeld V., Gelaki S., Nikshych D., Ostrik V., On braided fusion categories. I, Selecta Math. (N.S.) 16 (2010), 1-119, arXiv:0906.0620.
  18. Fairbairn W.M., Fulton T., Klink W.H., Finite and disconnected subgroups of SU3 and their application to the elementary-particle spectrum, J. Math. Phys. 5 (1964), 1038-1051.
  19. Feng B., Hanany A., He Y.-H., Prezas N., Discrete torsion, non-abelian orbifolds and the Schur multiplier, J. High Energy Phys. 2001 (2001), no. 1, 033, 25 pages, hep-th/0010023.
  20. Finch P.E., Dancer K.A., Isaac P.S., Links J., Solutions of the Yang-Baxter equation: descendants of the six-vertex model from the Drinfeld doubles of dihedral group algebras, Nuclear Phys. B 847 (2011), 387-412, arXiv:1003.0501.
  21. Flyvbjerg H., Character table for the 1080-element point-group-like subgroup of SU(3), J. Math. Phys. 26 (1985), 2985-2989.
  22. Gannon T., Modular data: the algebraic combinatorics of conformal field theory, math.QA/0103044.
  23. GAP - Groups, Algorithms, Programming - a system for computational discrete algebra,
  24. Ginsparg P., Curiosities at c=1, Nuclear Phys. B 295 (1988), 153-170.
  25. Grimus W., Ludl P.O., Principal series of finite subgroups of SU(3), arXiv:1006.0098.
  26. Groupprops, The group properties Wiki,
  27. Hayashi T., A canonical Tannaka duality for finite semisimple tensor categories, math.QA/9904073.
  28. Hoffman P.N., Humphrey J.F., Projective representations of the symmetric groups. Q-functions and shifted tableaux, Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1992.
  29. Hu X., Jing N., Cai W., Generalized McKay quivers of rank three, arXiv:1207.2823.
  30. Huang J.-S., Lectures on representation theory, World Scientific Publishing Co. Inc., River Edge, NJ, 1999.
  31. Kac V.G., Todorov I.T., Affine orbifolds and rational conformal field theory extensions of W1+∞, Comm. Math. Phys. 190 (1997), 57-111, hep-th/9612078.
  32. Koornwinder T.H., Schroers B.J., Slingerland J.K., Bais F.A., Fourier transform and the Verlinde formula for the quantum double of a finite group, J. Phys. A: Math. Gen. 32 (1999), 8539-8549, math.QA/9904029.
  33. Ludl P.O., On the finite subgroups of U(3) of order smaller than 512, J. Phys. A: Math. Theor. 43 (2010), 395204, 28 pages, Corrigendum, J. Phys. A: Math. Theor. 44 (2011), 139501, arXiv:1006.1479.
  34. Ludl P.O., Systematic analysis of finite family symmetry groups and their application to the lepton sector, arXiv:0907.5587.
  35. Luhn C., Nasri S., Ramond P., Tri-bimaximal neutrino mixing and the family symmetry Z7×Z3, Phys. Lett. B 652 (2007), 27-33, arXiv:0706.2341.
  36. Lusztig G., Exotic Fourier transform, Duke Math. J. 73 (1994), 227-241.
  37. Lusztig G., Unipotent representations of a finite Chevalley group of type E8, Quart. J. Math. Oxford Ser. (2) 30 (1979), 315-338.
  38. Mathematica, Version 9.0, Wolfram Research, Inc., Champaign, IL, 2012,
  39. McKay J., Graphs, singularities, and finite groups, in The Santa Cruz Conference on Finite Groups (Univ. California, Santa Cruz, Calif., 1979), Proc. Sympos. Pure Math., Vol. 37, Amer. Math. Soc., Providence, R.I., 1980, 183-186.
  40. Ocneanu A., Paths on Coxeter diagrams: from Platonic solids and singularities to minimal models and subfactors, Fields Institute Monographs, Amer. Math. Soc., Providence, RI, 1999.
  41. Ogievetsky O., Private communication, 1995.
  42. Ostrik V., Module categories over the Drinfeld double of a finite group, Int. Math. Res. Not. 2003 (2003), 1507-1520, math.QA/0202130.
  43. Ostrik V., Module categories, weak Hopf algebras and modular invariants, Transform. Groups 8 (2003), 177-206, math.QA/0111139.
  44. Parattu K.M., Wingerter A., Tribimaximal mixing from small groups, Phys. Rev. D 84 (2011), 013011, 17 pages, arXiv:1012.2842.
  45. Petkova V.B., Zuber J.-B., The many faces of Ocneanu cells, Nuclear Phys. B 603 (2001), 449-496, hep-th/0101151.
  46. Roan S.-S., Minimal resolutions of Gorenstein orbifolds in dimension three, Topology 35 (1996), 489-508.
  47. Schweigert C., Private communication.
  48. Thompson M., Towards topological quantum computation? - Knotting and fusing flux tubes, arXiv:1012.5432.
  49. Vafa C., Modular invariance and discrete torsion on orbifolds, Nuclear Phys. B 273 (1986), 592-606.
  50. Verlinde E., Fusion rules and modular transformations in 2D conformal field theory, Nuclear Phys. B 300 (1988), 360-376.
  51. Witherspoon S.J., The representation ring of the quantum double of a finite group, Ph.D. thesis, The University of Chicago, 1994, available at
  52. Yau S.S.-T., Yu Y., Gorenstein quotient singularities in dimension three, Mem. Amer. Math. Soc. 105 (1993), viii+88 pages.
  53. Yu J., A note on closed subgroups of compact Lie groups, arXiv:0912.4497.

Previous article  Next article   Contents of Volume 9 (2013)