On fusion categories


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.


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