Skip to main content
SpringerLink
Log in

On braided fusion categories I

  • Published:
Selecta Mathematica Aims and scope Submit manuscript

Abstract

We introduce a new notion of the core of a braided fusion category. It allows to separate the part of a braided fusion category that does not come from finite groups. We also give a comprehensive and self-contained exposition of the known results on braided fusion categories without assuming them pre-modular or non-degenerate. The guiding heuristic principle of our work is an analogy between braided fusion categories and Casimir Lie algebras.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Altschüler, D., Bruguières, A.: Handleslide for a sovereign category, preprint (2001)

  2. Arkhipov S., Gaitsgory D.: Another realization of the category of modules over the small quantum group. Adv. Math. 173(1), 114–143 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  3. Anderson G., Moore G.: Rationality in conformal field theory. Comm. Math. Phys. 117(3), 441–450 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  4. Barge J., Lannes J., Latour F., Vogel P.: Λ-spheres. Ann. Sci. École Norm. Suppl. 7(4), 463–505 (1974)

    MATH  MathSciNet  Google Scholar 

  5. Bezrukavnikov, R.: On tensor categories attached to cells in affine Weyl groups. In: Representation Theory of Algebraic Groups and Quantum Groups, Adv. Stud. Pure Math., pp 69–90, vol 40. Math. Soc. Japan, Tokyo (2004)

  6. Bruguières A.: Catégories prémodulaires, modularization et invariants des variétés de dimension 3. Math. Ann. 316(2), 215–236 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  7. Beilinson A., Drinfeld V.: Chiral Algebras. American Mathematical Society, Providence (2004)

    MATH  Google Scholar 

  8. Baues H.J., Conduché D.: The central series for Peiffer commutators in groups with operators. J. Algebra 133(1), 1–34 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  9. Bakalov, B., Kirillov, A. Jr.: Lectures on Tensor Categories and Modular Functors, AMS (2001)

  10. Carrasco P., Cegarra A.M.: Extensions centrales de Gr-catégories par des Gr-catégories tressées. C. R. Acad. Sci. Paris Sér. I Math. 321(5), 527–530 (1995)

    MATH  MathSciNet  Google Scholar 

  11. Conduché D., Ellis G.J.: Quelques propriétés homologiques des modules précroisés. J. Algebra 123(2), 327–335 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  12. Davydov, A., Müger, M., Nikshych, D., Ostrik, V.: Étale algebras in braided fusion categories, in preparation

  13. Deligne P.: Catégories tensorielles. Moscow Math. J. 2(2), 227–248 (2002)

    MATH  MathSciNet  Google Scholar 

  14. Drinfeld V.G.: Quasi-Hopf algebras. Leningrad Math. J. 1(6), 1419–1457 (1990)

    MathSciNet  Google Scholar 

  15. Etingof P., Nikshych D., Ostrik V.: On fusion categories. Ann. Math. 162, 581–642 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  16. Etingof P., Nikshych D., Ostrik V.: An analogue of Radford’s S 4 formula for finite tensor categories. Int. Math. Res. Notices 54, 2915–2933 (2004)

    Article  MathSciNet  Google Scholar 

  17. Etingof, P., Nikshych, D., Ostrik, V.: Weakly group-theoretical and solvable fusion categories, math.0809.3031

  18. Etingof, P., Nikshych, D., Ostrik, V.: Fusion categories and homotopy theory, preprint. arXiv:0909.3140

  19. Etingof P., Ostrik V.: Finite tensor categories. Moscow Math. J. 4, 627–654 (2004)

    MATH  MathSciNet  Google Scholar 

  20. Frenkel E., Gaitsgory D.: Geometric realizations of Wakimoto modules at the critical level. Duke Math. J. 143(1), 117–203 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  21. Gaitsgory, D.: The notion of category over an algebraic stack, arXiv:math/0507192

  22. Gelaki S., Nikshych D.: Nilpotent fusion categories. Adv. Math. 217(3), 1053–1071 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  23. Joyal A., Street R.: Braided tensor categories. Adv. Math. 102, 20–78 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  24. Kassel C.: Quantum Groups. Graduate Texts in Mathematics, vol 155. Springer, New York (1995)

    Google Scholar 

  25. Kath I., Olbrich M.: Metric Lie algebras and quadratic extensions. Transform. Groups 11(1), 87–131 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  26. Kirillov, A. Jr.: Modular categories and orbifold models II, arXiv:math/0110221

  27. Kirillov A. Jr, Ostrik V.: On a q-analogue of McKay correspondence and ADE classification of \({\hat{\mathfrak{s}\mathfrak{l}}_2}\) conformal field theories. Adv. Math. 171(2), 183–227 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  28. Lannes J.: Formes quadratiques d’enlacement sur l’anneau des entiers d’un corps de nombres. Ann. Sci. École Norm. Suppl. 8(4), 535–579 (1975)

    MATH  MathSciNet  Google Scholar 

  29. Milnor J., Husemoller D.: Symmetric Bilinear Forms, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 73. Springer, New York (1973)

    Google Scholar 

  30. Müger M.: Galois theory for braided tensor categories and the modular closure. Adv. Math. 150(2), 151–201 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  31. Müger M.: On the structure of modular categories. Proc. Lond. Math. Soc. 87, 291–308 (2003)

    Article  MATH  Google Scholar 

  32. Müger M.: Galois extensions of braided tensor categories and braided crossed G-categories. J. Algebra 277(1), 256–281 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  33. Müger M.: From subfactors to categories and topology. I. Frobenius algebras in and Morita equivalence of tensor categories. J. Pure Appl. Algebra 180(1–2), 81–157 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  34. Müger M.: From subfactors to categories and topology. II. The quantum double of tensor categories and subfactors. J. Pure Appl. Algebra 180(1–2), 159–219 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  35. Moore G., Seiberg N.: Classical and quantum conformal field theory. Commun. Math. Phys. 123, 177–254 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  36. Montgomery S., Witherspoon S.: Irreducible representations of crossed products. J. Pure Appl. Algebra 129(3), 315–326 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  37. Naidu D., Nikshych D.: Lagrangian subcategories and braided tensor equivalences of twisted quantum doubles of finite groups. Comm. Math. Phys. 279, 845–872 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  38. Ostrik V.: Module categories, weak Hopf algebras and modular invariants. Transform. Groups 8, 177–206 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  39. Ostrik V.: Module categories over the Drinfeld double of a finite group. Int. Math. Res. Not. 27, 1507–1520 (2003)

    Article  MathSciNet  Google Scholar 

  40. Ostrik V.: On formal codegrees of fusion categories. Math. Res. Lett. 16(5), 897–903 (2009)

    MathSciNet  Google Scholar 

  41. Pareigis B.: On braiding and dyslexia. J. Algebra 171(2), 413–425 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  42. Quinn, F.: Group categories and their field theories. In: Proceedings of the Kirbyfest (Berkeley, CA, 1998), pp. 407–453 (electronic), Geom. Topol. Monogr., 2, Geom. Topol. Publ., Coventry (1999)

  43. Serre J.-P.: A Course in Arithmetic, Graduate Texts in Mathematics, vol. 7. Springer, New York (1973)

    Google Scholar 

  44. Siehler, J.A.: Braided Near-group Categories, math.QA/0011037

  45. Tambara D., Yamagami S.: Tensor categories with fusion rules of self-duality for finite abelian groups. J. Algebra 209(2), 692–707 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  46. Turaev V.: Quantum invariants of knots and 3-manifolds. W. de Gruyter, Berlin (1994)

    MATH  Google Scholar 

  47. Turaev, V.: Homotopy field theory in dimension 3 and crossed group-categories, preprint. arXiv: math/0005291

  48. Vafa C.: Toward classification of conformal theories. Phys. Lett. B. 199, 195–202 (1987)

    Article  MathSciNet  Google Scholar 

  49. Wall C.T.C.: Quadratic forms on finite groups, and related topics. Topology 2, 281–298 (1963)

    Article  MATH  MathSciNet  Google Scholar 

  50. Weil A.: Sur certains groupes d’operateurs unitaires. Acta Math. 111, 143–211 (1964)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Chicago, Chicago, IL, 60637, USA

    Vladimir Drinfeld

  2. Department of Mathematics, Technion-Israel Institute of Technology, Haifa, 32000, Israel

    Shlomo Gelaki

  3. Department of Mathematics and Statistics, University of New Hampshire, Durham, NH, 03824, USA

    Dmitri Nikshych

  4. Department of Mathematics, University of Oregon, Eugene, OR, 97403, USA

    Victor Ostrik

Authors
  1. Vladimir Drinfeld
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shlomo Gelaki
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Dmitri Nikshych
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Victor Ostrik
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Victor Ostrik.

Additional information

To Yuri Ivanovich Manin with admiration.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Drinfeld, V., Gelaki, S., Nikshych, D. et al. On braided fusion categories I. Sel. Math. New Ser. 16, 1–119 (2010). https://doi.org/10.1007/s00029-010-0017-z

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00029-010-0017-z

Keywords

Mathematics Subject Classification (2000)

Search

Navigation