Classificaition of subfactors in operator algebra and its applications

我研究了一种方法，用辫子把较小的算子代数的自同态推广到较大的代数。这首先是由Longo和Rehren定义的，徐在一个略有不同的环境中进行了研究。另一方面，Ocneanu在与Longo-Rehren的环境完全不同的情况下，研究了与dykin图有关的手性投影仪的理论。Bockenhauer、Evans和我推广了α-诱导和手征射影的定义，并证明了它们给出了相同的构造。我们得到了模不变量和融合规则代数的几个结构结果。其次，我研究了S^1上的一个von Neumann代数网和它上的四个区间的子因子。我们证明了Xu的构造给出了一个与Longo-Rehren构造同构的子因子，并证明了编织的非简并性在这个条件下是自动成立的。我们根据手征分枝系数确定了由手征α-诱导产生的M-M融合规则代数的结构。作为应用，我们确定了与SU(2)_κ相关的所有模不变量以及与SU(3)_κ相关的共形包含产生的模不变量的完全M-M融合规则代数结构，并进一步研究了α-归纳法产生的Longo-Rehren子因子。我们可以描述由Longo-Rehren子因子产生的张量范畴。我们进一步证明，如果编织是非退化的，那么我们在α-诱导后得到的作为通常的Longo-Rehren子因子的对偶的子因子与由广义Longo-Rehren构造得到的子因子同构。

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.