On the center of fusion categories

TL;DR

This paper generalizes M"uger's 2003 result by proving that the center of a pivotal fusion category over an arbitrary commutative ring is 2-modular, extending the understanding of fusion categories beyond fields.

Contribution

It extends M"uger's theorem to pivotal fusion categories over any commutative ring, describing the Hopf algebra structure of the center and conditions for semisimplicity.

Findings

Center of C is 2-modular over any commutative ring.

Dimension of C is invertible iff objects are retracts of free half-braidings.

Center is semisimple iff dimension of C is non-zero over a field.

Abstract

M\"uger proved in 2003 that the center of a spherical fusion category C of non-zero dimension over an algebraically closed field is a modular fusion category whose dimension is the square of that of C. We generalize this theorem to a pivotal fusion category C over an arbitrary commutative ring K, without any condition on the dimension of the category. (In this generalized setting, modularity is understood as 2-modularity in the sense of Lyubashenko.) Our proof is based on an explicit description of the Hopf algebra structure of the coend of the center of C. Moreover we show that the dimension of C is invertible in K if and only if any object of the center of C is a retract of a `free' half-braiding. As a consequence, if K is a field, then the center of C is semisimple (as an abelian category) if and only if the dimension of C is non-zero. If in addition K is algebraically closed, then this condition implies that the center is a fusion category, so that we recover M\"uger's result.