Department of Mathematics,

University of New Hampshire,

Durham, NH 03824 USA

E-mail: nikshych@cisunix.unh.edu

**On fusion categories**

**Abstract:**

A fusion category is a semisimple tensor category with duality. Such
categories arise in several areas of mathematics and physics - conformal
field theory, operator algebras, representation theory of quantum groups,
and others. In this talk we present the results of our joint work with
Pavel Etingof
and Victor
Ostrik on the structure and properties of such categories. We show
that the global dimension of a fusion category is always positive and that
fusion categories and functors between them are undeformable (in particular,
the number of categories realizing a given fusion rule is finite). We also
develop the theory of Frobenius-Perron dimensions in a fusion category.