Classificaition of subfactors in operator algebra and its applications

点击翻译按钮获取中文摘要

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, studied by Xu in a slightly different setting. On the other hand, Ocneanu has studied theory of a chiral projector in connection to the Dynkin diagrams in a situation which looked entirely different from the setting of Longo-Rehren. Bockenhauer, Evans and I have extended definitions of both the α-induction and the chiral projector, and proved that they give the same construction. We have obatined several structure results for modular invariants and fusion rule algebras.Next I studied subfactors arising from a net of von Neumann algebras on S^1 and four intervals on it with Longo and Muger. We have proved that Xu's construction gives a subfactor isomorphic to the Longo-Rehren construction and prove that non-degeneracy of a braiding holds automatically in this setting.We have determined the structure of M-M fusion rule algebras arising from chiral α-induction using one braiding in terms of chiral branching coefficients. As applications, we have determined the full M-M fusion rule algebra structures for all modular invariants associated with SU(2)_κ and modular invariants arising from conformal inclusions associated with SU(3)_κ.We have further studied the Longo-Rehren subfactors arising from α-induction. We can describe the tensor categories arising from the Longo-Rehren subfactors. 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 α-induction is isomorphic to the one arising from the generalized Longo-Rehren construction.