tailieunhanh - Đề tài " 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. | Annals of Mathematics On fusion categories By Pavel Etingof Dmitri Nikshych and Viktor Ostrik Annals of Mathematics 162 2005 581 642 On fusion categories By Pavel Etingof Dmitri NiKSHycH and Viktor Ostrik 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. 1. Introduction Throughout this paper except for 9 k denotes an algebraically closed field of characteristic zero. By a fusion category C over k we mean a k-linear semisimple rigid tensor monoidal category with finitely many simple objects and finite dimensional spaces of morphisms such that the endomorphism algebra of the neutral object is k see BaKi . Fusion categories arise in several areas of mathematics and physics - conformal field theory operator algebras representation theory of quantum groups and others. This paper is devoted to the study of general properties of fusion categories. This has been an area of intensive research for a number of years and many remarkable results have been obtained. However many of these results were proved not in general but under various assumptions on the category. The goal of this paper is to remove such assumptions and to give an account of the theory of fusion categories in full .