We have studied alpha-induction, a method of extending an endomorphism from a sub-operator algebra to a larger operator algebra using braiding, as a continuation of the previous works.
In the previous year, we studied the "quantum doubles" of systems of endomorphisms arising from alpha-induction in [32]. In this study, we have shown that the same canonical endomorphism as for the generalized Longo-Rehren subfactor intrioduced by Rehren in 2000 can be also obtained by applying the usual Longo-Rehren construction after alpha-induction. This year, we have further shown that if the braiding is non-degenerate, then the subfactor we obtain as a dual of the usual Longo-Rehren subfactor after alpha-induction is isomorphic to the one arising from the generalized Longo-Rehren construction. If we restrict the extension of endomorphisms on one of the tensoring factors and use a chiral branching coefficient instead of a modular invariant, we still get a similar isomorphism theorem. We have also shown that the parent type I modular invariant of Bockenhauer-Evans can be also studied in the context of this construction.