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.

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The goal of this paper is to give a category theory based definition and classification of ''finite subgroups in U q ðsl 2 Þ'' where q ¼ e pi=l is a root of unity. We propose a definition of such a subgroup in terms of the category of representations of U q ðsl 2 Þ; we show that this definition is a natural generalization of the notion of a subgroup in a reductive group, and that it is also related with extensions of the chiral (vertex operator) algebra corresponding to b sl sl 2 at level k ¼ l À 2: We show that ''finite subgroups in U q ðsl 2 Þ'' are classified by Dynkin diagrams of types A n ; D 2n ; E 6 ; E 8 with Coxeter number equal to l; give a description of this correspondence similar to the classical McKay correspondence, and discuss relation with modular invariants in ð b sl sl 2 Þ k conformal field theory. The results we get are parallel to those known in the theory of von Neumann subfactors, but our proofs are independent of this theory. # 2002 Elsevier Science (USA)

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We classify semisimple rigid monoidal categories with two isomorphism classes of simple objects over the field of complex numbers. In the appendix written by P. Etingof it is proved that the number of semisimple Hopf algebras with a given finite number of irreducible representations is finite.