On the graded center of graded categories

TL;DR

This paper investigates the G-center of G-graded monoidal categories, proving that under certain conditions it forms a G-modular category, thus generalizing M"uger's theorem for the ungraded case.

Contribution

It extends M"uger's theorem by showing the G-center of a spherical G-fusion category is G-modular, broadening the understanding of graded monoidal categories.

Findings

G-center of a spherical G-fusion category is G-modular under specified conditions

Generalization of M"uger's theorem to G-graded categories

Identification of interesting objects within the G-center

Abstract

We study the G-centers of G-graded monoidal categories where G is an arbitrary group. We prove that for any spherical G-fusion category C over an algebraically closed field such that the dimension of the neutral component of C is non-zero, the G-center of C is a G-modular category. This generalizes a theorem of M. M\"uger corresponding to G=1. We also exhibit interesting objects of the G-center.