Longo–Rehren Subfactors Arising from $\alpha$-induction

Abstract

We study (dual) Longo–Rehren subfactors $M\otimes M^{\mathrm{opp}}\subset R$ arising from various systems of endomorphisms of $M$ obtained from $\alpha$-induction for some braided subfactor $N\subset M$. Our analysis provides useful tools to determine the systems of $R$-$R$ morphisms associated with such Longo–Rehren subfactors, which constitute the “quantum double” systems in an appropriate sense. The key to our analysis is that $\alpha$-induction produces half-braidings in the sense of Izumi, so that his general theory can be applied. Nevertheless, $\alpha$-induced systems are in general not braided, and thus our results allow to compute the quantum doubles of (certain) systems without braiding. We illustrate our general results by several examples, including the computation of the quantum double systems for the asymptotic inclusion of the E8 subfactor as well as its three analogues arising from conformal inclusions of $\text{SU}(3{)}_k$.