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On fusion categories

Pavel Etingof·Dmitri Nikshych·Viktor Ostrik
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Abstract

Using a variety of methods developed in the literature (in particular, the theory of weak Hopf algebras), we prove a number of general results about fusion categories in characteristic zero. We show that the global dimension of a fusion category is always positive, and that the S-matrix of any modular category (not necessarily hermitian) is unitary. We also show that the category of module functors between two module categories over a fusion category is semisimple, and that fusion categories and tensor functors between them are undeformable (generalized Ocneanu rigidity). In particular the number of such categories (functors) realizing a given fusion datum is finite. Finally, we develop the theory of Frobenius-Perron dimensions in an arbitrary fusion category and classify categories of prime dimension.

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Pavel Etingof, Dmitri Nikshych, Viktor Ostrik. 2002-03-07. On fusion categories. https://arxiv.org/abs/math/0203060

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