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.
