Algebras and Representation Theory

, Volume 21, Issue 6, pp 1353–1368 | Cite as

On the Classification of Almost Square-Free Modular Categories

  • Jingcheng DongEmail author
  • Sonia Natale


Let \(\mathcal {C}\) be a modular category of Frobenius-Perron dimension d q n , where q > 2 is a prime number and d is a square-free integer. We show that \(\mathcal {C}\) must be integral and nilpotent and therefore group-theoretical. In the case where q = 2, we describe the structure of \(\mathcal {C}\) in terms of equivariantizations of group-crossed braided fusion categories.


Braided fusion category Modular category Group-theoretical fusion category Braided G-crossed fusion category Tannakian category 

Mathematics Subject Classification (2010)

18D10 16T05 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The first author was partially supported by the Fundamental Research Funds for the Central Universities (KYZ201564), the Natural Science Foundation of China (11201231) and the Qing Lan Project. The second author was partially supported by CONICET and SeCYT–UNC.


  1. 1.
    Bruillard, P., Galindo, C., Hong, S.-M., Kashina, Y., et al.: Classification of integral modular categories of Frobenius-Perron dimension p q 4 and p 2 q 2. Canad. Math. Bull. 57(4), 721–734 (2014)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bruillard, P., Galindo, C., Ng, S.-H., Plavnik, J.Y., Rowell, E.C., Wang, Z.: On the classification of weakly integral modular categories. J. Pure Appl. Algebra 220(6), 2364–2388 (2016)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bruillard, P., Plavnik, J.Y., Rowell, E.C.: Modular categories of dimension p 3 m with m square-free. To appear in Proc. Amer. Math. Soc. arXiv:1609.04896
  4. 4.
    Burciu, S., Natale, S.: Fusion rules of equivariantizations of fusion categories. J. Math. Phys 54(1), 013511 (2013)MathSciNetCrossRefGoogle Scholar
  5. 5.
    de Wild Propitius, M.D.F.: Topological interactions in broken gauge theories, Ph. D. Thesis, University of Amsterdam. arXiv:hep-th/9511195 (1995)
  6. 6.
    Dijkgraaf, R., Pasquier, V., Roche, P.: Quasi-quantum groups related to Orbifold models. In: Proceedings of the Modern Quantum Field Theory, pp. 375–383. Tata Institute, Bombay (1990)Google Scholar
  7. 7.
    Dong, J.: Braided \(\mathbb {Z}_{q}\)-extensions of pointed fusion categories. Proc. Amer. Math. Soc. 145(3), 995–1001 (2017)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Dong, J., Li, L., Dai, L.: Integral almost square-free modular categories. J. Algebra Appl 16(5), 1750104, 14 (2017)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Dong, J., Tucker, H.: Integral modular categories of Frobenius-Perron dimension p q n. Algebr. Represent. Theor. 19, 33–46 (2016)CrossRefGoogle Scholar
  10. 10.
    Drinfeld, V., Gelaki, S., Nikshych, D., Ostrik, V.: Group-theoretical properties of nilpotent modular categories. arXiv:0704.0195
  11. 11.
    Drinfeld, V., Gelaki, S., Nikshych, D., Ostrik, V.: On braided fusion categories I. Selecta Math. New Ser. 16(1), 1–119 (2010)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Etingof, P., Gelaki, S.: Some properties of finite-dimensional semisimple Hopf algebras. Math. Res. Lett. 5(2), 191–197 (1998)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Etingof, P., Gelaki, S., Ostrik, V.: Classification of fusion categories of dimension p q. Int. Math. Res. Not. 2004(57), 3041–3056 (2004)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Etingof, P., Nikshych, D., Ostrik, V.: On fusion categories. Ann. Math. 162(2), 581–642 (2005)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Etingof, P., Nikshych, D., Ostrik, V.: Weakly group-theoretical and solvable fusion categories. Adv. Math. 226(1), 176–205 (2011)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Gelaki, S., Nikshych, D.: Nilpotent fusion categories. Adv. Math. 217(3), 1053–1071 (2008)MathSciNetCrossRefGoogle Scholar
  17. 17.
    González, M.E., Natale, S.: On fusion rules and solvability of a fusion category. J. Group Theory 20(1), 133–167 (2017)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Liptrap, J.: Generalized Tambara-Yamagami categories, preprint arXiv:1002.3166
  19. 19.
    Majid, S.: Quantum double for quasi-Hopf algebras. Lett. Math. Phys. 45, 1–9 (1998)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Mason, G., Ng, S.-H.: Group cohomology and gauge equivalence classes of some twisted quantum doubles. Trans. Amer. Math. Soc. 353, 3465–3509 (2001)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Müger, M.: On the structure of modular categories. Proc. London Math. Soc. 87(02), 291–308 (2003)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Müger, M.: Galois theory for braided tensor categories and the modular closure. Adv. Math. 150(2), 151–201 (2000)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Müger, M.: Galois extensions of braided tensor categories and braided crossed G-categories. J. Algebra 277(1), 256–281 (2004)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Naidu, D., Nikshych, D., Witherspoon, S.: Fusion subcategories of representation categories of twisted quantum doubles of finite groups . Int. Math. Res. Not. 2009(22), 4183–4219 (2009)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Naidu, D., Rowell, E.C.: A finiteness property for braided fusion categories. Algebr. Represent. Theory. 14(5), 837–855 (2011)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Natale, S.: On group theoretical Hopf algebras and exact factorizations of finite groups. J. Algebra 270(1), 199–211 (2003)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Natale, S.: Faithful simple objects, orders and gradings of fusion categories. Algebr. Geom. Topol. 13, 1489–1511 (2013)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Natale, S.: On weakly group-theoretical non-degenerate braided fusion categories. J. Noncommut. Geom. 8(4), 1043–1060 (2014)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2017

Authors and Affiliations

  1. 1.College of EngineeringNanjing Agricultural UniversityNanjingChina
  2. 2.Facultad de Matemática, Astronomía y FísicaUniversidad Nacional de Córdoba, CIEM – CONICET, Ciudad UniversitariaCórdobaArgentina

Personalised recommendations