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

  • 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 modular categories, fusion categories
MSC Classifications: 18D10 show english descriptions Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories [See also 19D23] 18D10 - Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories [See also 19D23]
 

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