On the center of fusion categories

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.