Fusion categories via string diagrams

Abstract

We use the string diagram calculus to give graphical proofs of the basic results of Etingof, Nikshych and Ostrik [On fusion categories, Ann. Math. 162 (2005) 581–642; arXiv:math/0203060, doi:10.4007/annals.2005.162.581] on fusion categories. These results include: the quadruple dual is canonically isomorphic to the identity, positivity of the paired dimensions, and Ocneanu rigidity. We introduce the pairing convention as a convenient graphical framework for working with fusion categories. We use this framework to express the pivotal operators as a product of the apex associator monodromy and the pivotal indicators. We also characterize pivotal structures as solutions of an explicit set of algebraic equations over the complex numbers, refining a formula of Wang.