Relative center construction for $G$-graded C$^*$-tensor categories and Longo-Rehren inclusions

TL;DR

This work provides an operator-algebraic realization of the relative center and the $G$-braiding for $G$-graded C$^*$-tensor categories via Longo-Rehren inclusions. By constructing a group action on the LR inclusion, it yields a $G$-braiding on the relative center and clarifies its relationship to the Drinfeld center of the full category through crossed products and Tube algebras. It also develops the $G$-equivariant center $Z^G(\\\mathcal{C})$ and the corresponding $G$-twisted Tube algebras, establishing 1-1 correspondences between irreducible objects and minimal central projections in the twisted tube framework. Overall, the results unify Gelaki–Naidu–Nikshych and Türaev–Virelizier perspectives within the LR paradigm, providing explicit braiding constructions and a coherent, operator-algebraic approach to centers under group actions.

Abstract

Gelaki-Naidu-Nikshych and Turaev-Virelizier showed the existence of $G$-braiding on the relative Drinfeld center of a $G$-graded tensor category. We will explain this concept from the viewpoint of Longo-Rehren inclusions.

Theorems & Definitions (38)

• Lemma 2.1
• Lemma 2.2
• Definition 2.3
• Proposition 2.4
• Definition 3.1
• Lemma 3.2
• Definition 3.3
• Proposition 3.4
• Lemma 3.5
• Lemma 3.6