Research abstract of Y. Kawahigashi for 1998-99

This year, I have studied a method to extend an endomorphism of a smaller operator algebra to a larger algebra, using a braiding. This was first defined by Longo and Rehren in 1994, studied by Xu in 1996 in a slightly different setting, and further studied extensively by Bockenhauer and Evans with the name "alpha-induction" since 1998. On the other hand, Ocneanu has studied theory of a chiral generator in connection to the Dynkin diagrams since 1994 in a situation which looked entirely different from the setting of Longo-Rehren. Bockenhauer, Evans and I have worked on the alpha-induction and the chiral generator in a general setting, and proved that they give the same construction. Furthermore, we have clarified their relations to modular invariants in conformal field theory by combining the two approaches. That is, if we start with a conformal inclusion, we can compute operator algebraic data from a modular invariant, and if we start with operator algebraic data, then we can compute a (combinatorial analogue of) modular invariant from them.

Next I studied subfactors constructed from a net of von Neumann algebras on S^1 and four intervals on it with Longo and Mueger. Xu has studied this construction for A. Wassermann's loop group construction, but its relation to the Longo-Rehren construction, an analogue of the quantum double construction, remained as a conjecture. We have proved that this four-interval construction gives a subfactor isomorphic to the Longo-Rehren construction in general and also that the braiding of the superselection sectors are automatically non-degenerate.

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