On fusion 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 (not necessarily hermitian) modular category 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. At the end of the paper we generalize some of these results to positive characteristic.

Authors

Pavel Etingof

Department of Mathematics
Massachusetts Institute of Technology
Cambridge, MA 02139
United States

Dmitri Nikshych

Department of Mathematics and Statistics
University of New Hampshire
Durham, NH 03824
United States

Viktor Ostrik

Department of Mathematics
University of Oregon
Eugene, OR 97403
United States