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Modular categories, integrality and Egyptian fractions


Authors: Paul Bruillard and Eric C. Rowell
Journal: Proc. Amer. Math. Soc. 140 (2012), 1141-1150
MSC (2010): Primary 18D10; Secondary 16T05, 11Y50
DOI: https://doi.org/10.1090/S0002-9939-2011-11476-X
Published electronically: December 8, 2011
MathSciNet review: 2869100
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Abstract: It is a well-known result of Etingof, Nikshych and Ostrik that there are finitely many inequivalent integral modular categories of any fixed rank $ n$. This follows from a double-exponential bound on the maximal denominator in an Egyptian fraction representation of $ 1$. A naïve computer search approach to the classification of rank $ n$ integral modular categories using this bound quickly overwhelms the computer's memory (for $ n\geq 7$). We use a modified strategy: find general conditions on modular categories that imply integrality and study the classification problem in these limited settings. The first such condition is that the order of the twist matrix is $ 2,3,4$ or $ 6$, and we obtain a fairly complete description of these classes of modular categories. The second condition is that the unit object is the only simple non-self-dual object, which is equivalent to odd-dimensionality. In this case we obtain a (linear) improvement on the bounds and employ number-theoretic techniques to obtain a classification for rank at most $ 11$ for odd-dimensional modular categories.


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Additional Information

Paul Bruillard
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
Email: paul.bruillard@math.tamu.edu

Eric C. Rowell
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
Email: rowell@math.tamu.edu

DOI: https://doi.org/10.1090/S0002-9939-2011-11476-X
Received by editor(s): December 3, 2010
Received by editor(s) in revised form: December 8, 2010
Published electronically: December 8, 2011
Communicated by: Harm Derksen
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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