On fusion categories
Pavel Etingof, Dmitri Nikshych, Viktor Ostrik
TL;DR
This work establishes foundational, characteristic-zero results for fusion categories via weak Hopf algebras, proving global dimension positivity, unitary S-matrices for modular categories, and Ocneanu rigidity (no deformations). It develops a comprehensive framework connecting fusion categories, weak Hopf algebras, and Yetter cohomology to prove semisimplicity of module-functor categories and finiteness of realizations of fusion data. The Frobenius-Perron theory is systematically integrated, yielding structure theorems for FP-dimensions, cyclotomicity, and group-theoretical classifications, and offering robust lifting arguments to positive characteristic. Together, these results provide a cohesive arithmetic and categorical picture of fusion categories, their centers, duals, and module categories, with broad implications for classification and rigidity phenomena in tensor categories.
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.