I have continued the joint research with D. E. Evans and J. Bockenhauer on alpha-induction which is a method to extend an endomorphism of a smaller algebra N to a larger operator algebra M using a braiding.
First, we have determined the structure of M-M fusion rule algebras arising from chiral alpha-induction using one braiding in terms of chiral branching coefficients. This is a continuation of our work in the previous year on the structure of M-M fusion rule algebras arising from mixed alpha-induction. As a by-product, we have proved non-degeneracy of the braiding on the ambichiral system. As applications, we have determined the full M-M fusion rule algebra structures for all modular invariants associated with SU(2)_k and modular invariants arising from conformal inclusions associated with SU(3)_k.
We have further studied the Longo-Rehren subfactors arising from alpha-induction. The key observation is that a relative braiding studied by Bockenhauer-Evans gives a half-braiding in the sense of Izumi. With this result, we can describe the tensor categories arising from the Longo-Rehren subfactors, which can be regarded as operator algebraic realization of "quantum doubles". In particular, we get a simpler proof and more detailed analysis for Rehren's new construction, a generalization of (a part of) Izumi's computations, a generalization of Ocneanu's announcements, and computations of fusion rule algebras associated with Longo-Rehren subfactors arising from conformal inclusions associated with SU(3)_k.