The Relative Drinfeld Commutant of a Fusion Category and α-Induction

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 α-induction for braided subfactors. We present examples arising from chiral conformal field theory. make analogous computations for the relative commutants of these fusion categories arising from α-induction. Our methods are similar to those in [7] and rely on half-braidings arising from relative braidings studied in [4].We refer the reader to [11] for a general theory of subfactors and to [15] for a review on subfactor theory, category theory, and conformal field theory.