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 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.

50 pages, latex; a reference to Schauenburg's freeness theorem was added and a new section 5.11 to fill a gap in the previous version; also Example 7.2 was corrected; in April 2017 a comment added in Subsection 9.3 giving a reference which fills a gap in this subsection

References in corpus (4)

• Spherical Categories
• Finite Quantum Groupoids and Their Applications
• On Higher Frobenius-Schur Indicators
• Doubles of Quasi-Quantum Groups

Cited by in corpus (7)

• Towards a mathematical formalism for classifying phases of matter
• Cyclic extensions of fusion categories via the Brauer-Picard groupoid
• Symmetry enhancement in RCFT
• Rank 4 Premodular Categories
• An analogue of Radford's S^4 formula for finite tensor categories
• Modular Categories Associated to Unipotent Groups
• Semisimple weak Hopf algebras