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# Classification of Integral Modular Categories of Frobenius--Perron Dimension $pq^4$ and $p^2q^2$

Published:2014-02-10
Printed: Dec 2014
• Paul Bruillard,
Department of Mathematics, Texas A$\&$M University, College Station, Texas 77843, USA
• César Galindo,
Departamento de matemáticas, Universidad de los Andes, Bogotá, Colombia
• Seung-Moon Hong,
Department of Mathematics and Statistics, University of Toledo, Ohio 43606, USA
• Yevgenia Kashina,
Department of Mathematical Sciences, DePaul University, Chicago, Illinois 60614, USA
• Deepak Naidu,
Department of Mathematical Sciences, Northern Illinois Universit, DeKalb, Illinois 60115, USA
• Sonia Natale,
Facultad de Matemática, Astronomía y Física, Universidad Nacional de Córdoba, CIEM--CONICET, Córdoba, Argentina
• Julia Yael Plavnik,
Facultad de Matemática, Astronomía y Física, Universidad Nacional de Córdoba, CIEM--CONICET, Córdoba, Argentina
• Eric C. Rowell,
Department of Mathematics, Texas A$\&$M University, College Station, Texas 77843, USA
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## Abstract

We classify integral modular categories of dimension $pq^4$ and $p^2q^2$, where $p$ and $q$ are distinct primes. We show that such categories are always group-theoretical except for categories of dimension $4q^2$. 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-group-theoretical integral modular category of dimension $4q^2$ is equivalent to either one of these well-known examples or is of dimension $36$ and is twist-equivalent to fusion categories arising from a certain quantum group.
 Keywords: modular categories, fusion categories
 MSC Classifications: 18D10 - Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories [See also 19D23]