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. Results on squared norms, global dimensions, weak Hopf algebras, and Ocneanu rigidityLet k be an algebraically closed field.By a multi-fusion category over k we mean a rigid semisimple k-linear tensor category C with finitely many simple objects and finite dimensional spaces of morphisms.If the unit object 1 of

Locations

• Annals of Mathematics - View - PDF