# The relative Drinfeld commutant of a fusion category and $\alpha$-induction

## TL;DR

This paper explores the structure of relative Drinfeld commutants in fusion categories, establishing a correspondence with half-braidings and central projections, and applies these results to categories from $lpha$-induction and conformal field theory.

## Contribution

It introduces a new correspondence linking relative commutants, half-braidings, and central projections, and computes examples from $lpha$-induction in conformal field theory.

## Key findings

- Established a correspondence among simple objects, half-braidings, and central projections.
- Explicitly computed relative Drinfeld commutants for categories from $lpha$-induction.
- Presented examples from chiral conformal field theory.

## Abstract

We establish a correspondence among simple objects of the relative commutant of a full fusion subcategory in a larger fusion category in the sense of Drinfeld, irreducible half-braidings of objects in the larger fusion category with respect to the fusion subcategory, and minimal central projections in the relative tube algebra. Based on this, we explicitly compute certain relative Drinfeld commutants of fusion categories arising from $\alpha$-induction for braided subfactors. We present examples arising from chiral conformal field theory.