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Classification of Integral Modular Categories of Frobenius–Perron Dimension pq4 and p2q2

Published online by Cambridge University Press:  20 November 2018

Paul Bruillard ,
Cásar Galindo ,
Seung-Moon Hong ,
Yevgenia Kashina ,
Deepak Naidu ,
Sonia Natale ,
Julia Yael Plavnik  and
Eric C. Rowell
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.arplavnik@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.arplavnik@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.

Keywords

18D10modular categoriesfusion categories
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 57 , Issue 4 , 01 December 2014 , pp. 721 - 734
Copyright
Copyright © Canadian Mathematical Society 2014

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