Skip to main content
SpringerLink
Log in

Twisted K-Theory and Modular Invariants: I Quantum Doubles of Finite Groups

  • Conference paper
Operator Algebras

Part of the book series: Abel Symposia ((ABEL,volume 1))

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Altschuler, A., Coste, A.: Quasi-quantum groups, knots, three manifolds and topological field theory. Commun. Math. Phys., 150, 83-107 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  2. Altschuler, A., Coste, A., Maillard, J-M.: Representation theory of twisted group double. Ann. Fond. Louis de Broglie. 29, 681-694 (2004).

    MathSciNet  Google Scholar 

  3. Atiyah, M.F.: K-theory past and present. Sitzungsberichte der Berliner Mathematischen Gesellschaft, 411-417, Berliner Math. Gesellschaft, Berlin, 2001. math.KT/0012213

    Google Scholar 

  4. Atiyah, M.F., Segal, G.B.: Twisted K -theory. math.KT/0407054

    Google Scholar 

  5. Bakalov, B., Kirillov, A.: Lectures on tensor cateogries and modular functors. University Lecture Series, vol 21, American Mathematical Society, Providence, R.I. 2001

    Google Scholar 

  6. Baumgärtel, H.: Operatoralgebraic methods in quantum field theory. Berlin: Akademie Verlag, 1995

    MATH  Google Scholar 

  7. Böckenhauer, J., Evans, D.E.: Modular invariants, graphs and α-induction for nets of subfactors. I. Commun. Math. Phys., 197, 361-386 (1998); II Commun. Math. Phys., 200, 57-103 (1999); III Commun. Math. Phys. 205, 183-228 (1999)

    Google Scholar 

  8. Böckenhauer, J., Evans, D.E.: Modular invariants from subfactors: Type I coupling matrices and intermediate subfactors. Commun. Math. Phys., 213, 267-289 (2000)

    Article  MATH  Google Scholar 

  9. Böckenhauer, J., Evans, D.E.: Modular invariants from subfactors, in Quantum symmetries in theoretical physics and mathematics (Bariloche, 2000), pp 95-131, Contemp. Math., 294, Amer. Math. Soc., Providence, RI, 2002

    Google Scholar 

  10. Böckenhauer, J., Evans, D.E.: Modular invariants and subfactors. Mathematical physics in mathematics and physics (Siena, 2000), 11-37, Fields Inst. Commun., 30, Amer. Math. Soc., 2001.

    Google Scholar 

  11. Böckenhauer, J., Evans, D.E., Kawahigashi, Y.: On α-induction, chiral generators and modular invariants for subfactors. Commun. Math. Phys., 208, 429-487 (1999)

    Article  MATH  Google Scholar 

  12. Böckenhauer, J., Evans, D.E., Kawahigashi, Y.: Chiral structure of modular invariants for subfactors. Commun. Math. Phys., 210, 733-784 (2000)

    Article  MATH  Google Scholar 

  13. Böckenhauer, J., Evans, D.E., Kawahigashi, Y.: Longo-Rehren subfactors arising from α-induction. Publ. RIMS, Kyoto Univ., 37, 1-35 (2001)

    Article  MATH  Google Scholar 

  14. Bouwknegt, P., Carey, A.L., Mathai, V., Murray, M.M., Stevenson, D.: Twisted K-theory and K-theory of bundle gerbes. hep-th/0106194

    Google Scholar 

  15. Cardy, J.: Fusion rules and the Verlinde formula. Nucl. Phys. B, 324, 581-596 (1989)

    Article  MathSciNet  Google Scholar 

  16. Coste, A., Gannon, T., Philippe, R.: Finite group modular data. Nucl. Phys. B, 581,679-717 (2000)

    Article  MATH  Google Scholar 

  17. Dijkgraaf, R.; Cumrun, V.; Verlinde, E.; Verlinde, H.: The operator algebra of orbifold models. Commun. Math. Phys., 123, 485-526 (1989)

    Article  MATH  Google Scholar 

  18. Dijkgraaf, R., Pasquier, V., Roche, P.: Quasi-quantum groups related to orbifolds models. Modern quantum field theory (Bombay, 1990), 375-383, World Sci. Publishing, River Edge, NJ, 1991.

    Google Scholar 

  19. Dixmier, J., Douady, A.: Champs continus d’espaces hilbertiens et de C*algebras. Bull. Soc. Math. France, 91, 227-284 (1963)

    MathSciNet  MATH  Google Scholar 

  20. Evans, D.E.: Fusion rules of modular invariants. Rev. Math. Phys., 14, 709-732 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  21. Evans, D.E.: Critical phenomena, modular invariants and operator algebras. Operator algebras and mathematical physics (Constanţa 2001, J. Cuntz, G.A. Elliott, S. Stratila et al. eds.), The Theta Foundation, Bucharest, 2003

    Google Scholar 

  22. Evans, D.E.: Modular invariant partition functions in statistical mechanics, conformal field theory and their realisation by subfactors. Proceedings Congress of International Association of Mathematical Physics (Lisbon 2003, J-C Zambrini, eds.), pp. 464-475, World Scientific Press. Singapore, 2005.

    Google Scholar 

  23. Evans, D. E., Kawahigashi, Y.: Quantum symmetries on operator algebras. Oxford University Press (1998)

    Google Scholar 

  24. Evans, D.E., Pinto, P.R.: Modular invariants and their fusion rules in Advances in Quantum Dynamics (Mount Holyoke, 2002), pp. 119-130, Contemp. Math. 335, Amer. Math. Soc., Providence, RI, 2003

    Google Scholar 

  25. Evans, D.E., Pinto, P.R.: Subfactor realisation of modular invariants. Commun. Math. Phys., 237, 309-363 (2003)

    MathSciNet  MATH  Google Scholar 

  26. Evans, D.E., Pinto, P.R.: Modular invariants and the double of the Haagerup subfactor. In Advances in Operator Algebras and Mathematical Physics (Sinaia 2003, F.-P. Boca, O. Bratteli, R. Longo, H. Siedentop eds.), pp. 67-88, The Theta Foundation, Bucharest, 2006.

    Google Scholar 

  27. Evans, D.E., Pinto, P.R.: Subfactor realisation of modular invariants, II.

    Google Scholar 

  28. Freed, D.S.: Higher algebraic structures and quantisation. Commun. Math. Phys., 159, 343-398 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  29. Freed, D.S.: The Verlinde algebra is twisted equivariant K-theory. Turkish J. Math., 25, 159-167 (2001).

    MathSciNet  MATH  Google Scholar 

  30. Freed, D.S.: Twisted K-theory and loop groups. Proceedings of the International Congress of Mathematicians, Vol. III (Beijing, 2002), 419-430, Higher Ed. Press, Beijing, 2002.

    Google Scholar 

  31. Freed, D.S.: K-theory in quantum field theory. Current developments in mathematics, 2001, 41-87, Int. Press, Somerville, MA, 2002.

    Google Scholar 

  32. Freed, D.S., Hopkins, M.J., Teleman, C.: Twisted equivariant K-theory with complex coefficients. math-ph/020631

    Google Scholar 

  33. Freed, D.S., Quinn, F.: Chern Simmons theory with finite gauge group. Commun. Math. Phys., 156, 435-472 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  34. di Francesco, P., Zuber, J.B.: SU (N ) lattice integrable models associated with graphs. Nucl. Phys. B, 338, 602-646 (1990)

    Article  MathSciNet  Google Scholar 

  35. Fröhlich, J., Gabbiani, F.: Operator algebras and conformal field theory. Commun. Math. Phys., 155, 569-640 (1993)

    Article  MATH  Google Scholar 

  36. Gannon, T.: Modular data: the algebraic combinatorics of conformal field theory. hep-th/0103044

    Google Scholar 

  37. Izumi, M.: The structure of sectors associated with Longo-Rehren inclusions, I. General theory. Commun. Math. Phys., 213, 127-179 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  38. Izumi, M.: The structure of sectors associated with Longo-Rehren inclusions, II. Examples. Rev. Math. Phys., 13, 603-674 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  39. Izumi, M., Kosaki, H.: On the subfactor analogue of the second cohomology. Preprint, Kyoto-Math 2002-01

    Google Scholar 

  40. Jones, V. F. R.: An invariant for group actions. Algèbres d’opèrateurs (Sèm., Les Plans-sur-Bex, 1978), pp. 237-253, Lecture Notes in Math., 725, Springer, Berlin, 1979.

    Google Scholar 

  41. Jones, V.F.R.: Index for subfactors, Invent. Math., 72, 1-25 (1983).

    Article  MathSciNet  MATH  Google Scholar 

  42. Katayama, Y., Takesaki, M.: Outer actions of a countable discrete amenable group on an AFD factor. Advances in quantum dynamics (South Hadley, MA, 2002),163-171, Contemp. Math., 335, Amer. Math. Soc., Providence, RI, 2003.

    Google Scholar 

  43. Kawahigashi, Y., Sato, N., Wakui, M. (2+1)-dimensional topological quantum field theory from subfactors and Dehn surgery formula for 3-manifold invariants. Preprint math.OA/0208238

    Google Scholar 

  44. Kirillov, A., Jr.; Ostrik, V.: On a q-analogue of the McKay correspondence and the ADE classification of sl2 conformal field theories. Adv. Math., 171, 183-227 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  45. Kosaki, H., Yamagami, S.: Irreducible bimodules associated with crossed product algebras. Internat. J. Math., 3, 661-676 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  46. Kosaki, H., Munemasa, A., Yamagami, S.: On fusion algebras associated to finite group actions. Pacific J. Math., 177, 269-290 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  47. Longo, R.: Index of subfactors and statistics of quantum fields, II. Commun. Math. Phys., 130, 285-309 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  48. Longo, R., Rehren, K.-H.: Nets of subfactors. Rev. Math. Phys., 7, 567-597 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  49. Mason, G.: The quantum double of a finite group and its role in conformal field theory. Groups ’93 Galway/St. Andrews, Vol. 2, 405-417, London Math. Soc. Lecture Note Ser., 212, Cambridge Univ. Press, Cambridge, 1995

    Chapter  Google Scholar 

  50. Ocneanu, A.: Paths on Coxeter diagrams: From Platonic solids and singularities to minimal models and subfactors, in Lectures on operator theory, Rajarama Bhat, B.V. et al. (eds.) (Notes recorded by S. Goto), The Fields Institute Monographs, Providence, Rhode Island: AMS publications 2000, pp. 243-323

    Google Scholar 

  51. Ocneanu, A.: The classification of subgroups of quantum SU (N ), in Quantum symmetries in theoretical physics and mathematics (Bariloche, 2000), Contemp. Math., 294, Amer. Math. Soc., Providence, RI, 2002, pp 133-159

    Google Scholar 

  52. Ostrik, V.: Module categories, weak Hopf algebras and modular invariants. Transform. Groups, 8, 177- 206 (2003) or math.QA/0111139

    Article  MathSciNet  MATH  Google Scholar 

  53. Ostrik, V.: Module categories for quantum doubles of finite groups. Inter. Math. Research. Notices, 27, 1507-1520 (2003), math.QA/0202130

    Article  MathSciNet  Google Scholar 

  54. Rieffel, M.: Morita equivalence for operator algebras. Operator algebras and applications, Part I (Kingston, Ont., 1980), pp. 285-298, Proc. Sympos. Pure Math., 38, Amer. Math. Soc., Providence, R.I., 1982.

    Google Scholar 

  55. Sato, N., Wakui, M.: Computations of Turaev-Viro-Ocneanu invariants of 3-manifolds from subfactors. J. Knot Theory Ramifications, 12, 543-574 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  56. Sutherland, C.: Cohomology and extensions of von Neumann algebras. I, II. Publ. RIMS, Kyoto Univ., 16, 105-133, 135-174 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  57. Turaev, V.G. Quantum invariants of knots and 3-manifolds. in de Gruyter Studies in Mathematics, vol 18. Walter de Gruyter, Berlin, 1994

    Google Scholar 

  58. Verlinde, E.: Fusion rules and modular transformations in 2D conformal field theory. Nucl. Phys. B, 300, 360-376 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  59. Wassermann, A.: Operator algebras and conformal field theory III: fusion of positive representations of LSU(n) using bounded operators. Invent. Math., 33, 467-538 (1998)

    Article  MathSciNet  Google Scholar 

  60. Wassermann, A.: Automorphic represenations of real and p-adic Lie groups. Noncommutative Geometry, Oberwolfach Report No. 45/2004.

    Google Scholar 

  61. Xu, F.: New braided endomorphisms from conformal inclusions. Commun. Math. Phys., 192, 349-403 (1998)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, Cardiff University, Senghennydd Road, CF24 4YG, Cardiff, Wales, UK

    David E Evans

Authors
  1. David E Evans
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Oslo, PB 1053 - Blindern, 0316, Oslo, Norway

    Ola Bratteli

  2. Department of Mathematics, University of Oslo, PB 1053 - Blindern, 0316, Oslo, Norway

    Sergey Neshveyev

  3. Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491, Trondheim, Norway

    Christian Skau

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Evans, D.E. (2006). Twisted K-Theory and Modular Invariants: I Quantum Doubles of Finite Groups. In: Bratteli, O., Neshveyev, S., Skau, C. (eds) Operator Algebras. Abel Symposia, vol 1. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-34197-0_6

Download citation

Publish with us

Policies and ethics

Search

Navigation