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Classification of Integral Modular Categories of Frobenius–Perron Dimension pq 4 and p 2 q 2

Published online by Cambridge University Press:  20 November 2018

Paul Bruillard
Affiliation:
Department of Mathematics, Texas A&M University, College Station, Texas 77843, USA e-mail: pjb2357@gmail.com
Cásar Galindo
Affiliation:
Departamento de Matemáticas, Universidad de los Andes, Bogotá, Colombia e-mail: cn.galindo1116@uniandes.edu.co
Seung-Moon Hong
Affiliation:
Department of Mathematics and Statistics, University of Toledo, Ohio 43606, USA e-mail: seungmoon.hong@utoledo.edu
Yevgenia Kashina
Affiliation:
Department of Mathematical Sciences, DePaul University, Chicago, Illinois 60614, USA e-mail: ykashina@depaul.edu
Deepak Naidu
Affiliation:
Department of Mathematical Sciences, Northern Illinois University, DeKalb, Illinois 60115, USA e-mail: dnaidu@math.niu.edu
Sonia Natale
Affiliation:
Facultad de Matemática, Astronomía y Física, Universidad Nacional de Córdoba, CIEM–CONICET, Córdoba, Argentina e-mail: natale@famaf.unc.edu.ar plavnik@famaf.unc.edu.ar
Julia Yael Plavnik
Affiliation:
Facultad de Matemática, Astronomía y Física, Universidad Nacional de Córdoba, CIEM–CONICET, Córdoba, Argentina e-mail: natale@famaf.unc.edu.ar plavnik@famaf.unc.edu.ar
Eric C. Rowell
Affiliation:
Department of Mathematics, Texas A & M University, College Station, Texas 77843, USA e-mail: rowell@math.tamu.edu
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Abstract

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We classify integral modular categories of dimension $p{{q}^{4}}$ and ${{p}^{2}}{{q}^{2}}$ , where $p$ and $q$ are distinct primes. We show that such categories are always group-theoretical, except for categories of dimension $4{{q}^{2}}$ . In these cases there are well-known examples of non-group-theoretical categories, coming from centers of Tambara–Yamagami categories and quantum groups. We show that a non-grouptheoretical integral modular category of dimension $4{{q}^{2}}$ is either equivalent to one of these well-known examples or is of dimension 36 and is twist-equivalent to fusion categories arising froma certain quantum group.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2014

References

[DMNO] Davydov, A., M. M¨uger, Nikshych, D., and Ostrik, V., The Witt group of non-degenerate braided fusion categories.J. Reine Angew. Math. 677 (2013), 135177.Google Scholar
[Ng] Dong, C., Lin, X., and Ng, S., Congruence property and galois symmetry of modular categories. arxiv:1201.6644.Google Scholar
[DGNO1] Drinfeld, V., Gelaki, S., Nikshych, D., and Ostrik, V., Group-theoretical properties of nilpotent modular categories. arxiv:0704.0195.Google Scholar
[DGNO2] Drinfeld, V., Gelaki, S., Nikshych, D., and Ostrik, V., On braided fusion categories. I. Selecta Math. (N. S.) 16 (2010), no. 1, 1119. http://dx.doi.org/10.1007/s00029-010-0017-z CrossRefGoogle Scholar
[EG] Etingof, P. and Gelaki, S., Some properties of finite-dimensional semisimple Hopf algebras.Math. Res. Lett. 5 (1998), no. 12, 191197. http://dx.doi.org/10.4310/MRL.1998.v5.n2.a5 CrossRefGoogle Scholar
[EGO] Etingof, P., Gelaki, S., and Ostrik, V., Classification of fusion categories of dimension pq.Int.Math. Res. Not. 2004, no. 57, 30413056.CrossRefGoogle Scholar
[ENO1] Etingof, P., Nikshych, D., and Ostrik, V., On fusion categories.Ann. of Math. (2) 162 (2005), no. 2, 581642. http://dx.doi.org/10.4007/annals.2005.162.581 CrossRefGoogle Scholar
[ENO2] Etingof, P., Nikshych, D., and Ostrik, V., Weakly group-theoretical and solvable fusion categories.Adv. Math. 226 (2011), no. 1, 176205. http://dx.doi.org/10.1016/j.aim.2010.06.009 CrossRefGoogle Scholar
[ENO3] Etingof, P., Nikshych, D., and Ostrik, V., Fusion categories and homotopy theory.Quantum Topol. 1 (2010), no. 3, 209273. http://dx.doi.org/10.4171/QT/6 CrossRefGoogle Scholar
[ERW] Etingof, P., Rowell, E. C., and S. J.Witherspoon, Braid group representations from twisted quantum doubles of finite groups.Pacific J. Math. 234 (2008), no. 1, 3341. http://dx.doi.org/10.2140/pjm.2008.234.33 CrossRefGoogle Scholar
[G1] Galindo, C., Clifford theory for tensor categories.J. London Math. Soc. (2) 83 (2011), no. 1, 5778. http://dx.doi.org/10.1112/jlms/jdq064 CrossRefGoogle Scholar
[G2] Galindo, C., Clifford theory for graded fusion categories.Israel J. Math. 192 (2012), no. 2, 841867. http://dx.doi.org/10.1007/s11856-012-0055-7 CrossRefGoogle Scholar
[GHR] Galindo, C., Hong, S.-M., and Rowell, E. C., Generalized and quasi-localizations of braid group representations.Int. Math. Res. Not. 2013, no. 3, 693731.CrossRefGoogle Scholar
[GNN] Gelaki, S., Naidu, D., and Nikshych, D., Centers of graded fusion categories.Algebra Number Theory 3 (2009), no. 8, 959990. http://dx.doi.org/10.2140/ant.2009.3.959 CrossRefGoogle Scholar
[GN] Gelaki, S. and Nikshych, D., Nilpotent fusion categories.Adv. Math. 217 (2008), no. 3, 10531071. http://dx.doi.org/10.1016/j.aim.2007.08.001 CrossRefGoogle Scholar
[JL] Jordan, D. and Larson, E., On the classification of certain fusion categories. J. Noncommut. Geom. 3 (2009), no. 3, 481499. http://dx.doi.org/10.4171/JNCG/44 CrossRefGoogle Scholar
[KW] Kazhdan, D. and H.Wenzl, Reconstructing monoidal categories. In: I. M. Gelfand Seminar, Adv. Soviet Math., 16, Part 2, American Mathematical Society, Providence, RI, 1993, pp. 111136.Google Scholar
[K] Kirillov, A., Jr., Modular categories and orbifold models. II. arxiv:math/0110221.Google Scholar
[M1] Müger, M., Galois theory for braided tensor categories and the modular closure.Adv. Math. 150 (2000), no. 2, 151201. http://dx.doi.org/10.1006/aima.1999.1860 CrossRefGoogle Scholar
[M2] Müger, M., On the structure of modular categories.Proc. London Math. Soc. 87 (2003), no. 2, 291308. http://dx.doi.org/10.1112/S0024611503014187 CrossRefGoogle Scholar
[M3] Müger, M., Galois extensions of braided tensor categories and braided crossed G-categories.J. Algebra 277 (2004), no. 1, 256281. http://dx.doi.org/10.1016/j.jalgebra.2004.02.026 CrossRefGoogle Scholar
[NNW] Naidu, D., Nikshych, D., and S.Witherspoon, Fusion subcategories of representation categories of twisted quantum doubles of finite groups.Int. Math. Res. Not. 2009, no. 22, 41834219.Google Scholar
[NR] Naidu, D. and Rowell, E. C., A finiteness property for braided fusion categories.Algebr. Represent. Theory. 14 (2011), no. 5, 837855. http://dx.doi.org/10.1007/s10468-010-9219-5 CrossRefGoogle Scholar
[Na1] Natale, S., On group theoretical Hopf algebras and exact factorizations of finite groups.J. Algebra 270 (2003), no. 1, 199211. http://dx.doi.org/10.1016/S0021-8693(03)00464-2 CrossRefGoogle Scholar
[Na2] Natale, S., On weakly group-theoretical non-degenerate braided fusion categories. arxiv:1301.6078.Google Scholar
[O] Ostrik, V., Module categories over the Drinfeld double of a finite group.Int. Math. Res. Not. 2003, no. 27, 15071520.CrossRefGoogle Scholar
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