Classificaition of subfactors in operator algebra and its applications

算子代数子因子的分类及其应用

• 批准号: 10640200
• 负责人: KAWAHIGASHI Yasuyuki
• 金额: $ 2.05万
• 依托单位: University of Tokyo
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1998
• 资助国家: 日本
• 起止时间: 1998 至 2000
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640200/
• 关键词: operator algebra; subfactor; conformal field theory; alpha-inibuction; quantum double; subfactor; Longo; induction; modular invariant; conformal field; Gaiois correspondence; paragroup; 部分因子環; Dynkin 図形; conformal inclusion; loop group

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.

研究了利用辫子将一个小算子代数的自同态推广到一个大算子代数的方法。这一概念最早由Longo和Luberren定义，由Xu在一个稍微不同的环境中进行研究。另一方面，Ocneanu研究了理论的手征投影机连接到Dynkin图的情况下，看起来完全不同的设置隆戈-Escherren。Bockenhauer，Evans和我推广了α-诱导和手征投影的定义，并证明了它们给出了相同的结构。我们已经得到了模不变量和融合规则代数的几个结构结果。接下来，我们利用Longo和Muger研究了S^1上的vonNeumann代数网及其上的四个区间所产生的子因子。我们证明了Xu构造给出了一个同构于Longo-Wehren构造的子因子，证明了在此情形下辫子的非退化性自动成立，并利用手征分支系数确定了由手征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SU（2）_κ模不变量和SU（3）_κ共形包含模不变量的完全M-M融合规则代数结构，并进一步研究了由α-归纳产生的Longo-Escherren子因子.我们可以描述由Longo-Escherren子因子产生的张量范畴。我们进一步证明了，如果辫子是非退化的，那么我们在α-归纳后得到的作为通常的Longo-Escherren子因子的对偶的子因子同构于由广义Longo-Escherren构造产生的子因子.

项目成果

J.Bockanlaner D.E.Evans.Y.Kawahigashi: "On α-induction, chiral projectors and modular invariants for subfactors"Commun.Math.Phys.. 208. 429-487 (1999)

J.Bockanlaner D.E.Evans.Y.Kawahigashi：“关于 α 归纳、手征投影仪和子因子的模不变量”Commun.Math.Phys.. 208. 429-487 (1999)

J.BSckarhaner D.E.Evans C.Kawahigashi: "Chirol Structure of modular invaviants for subfactors"Commun.Math.Phys.. 210. 733-784 (2000)

J.BSckarhaner D.E.Evans C.Kawahigashi：“子因子的模不变量的 Chirol 结构”Commun.Math.Phys.. 210. 733-784 (2000)

J.Bockenhauer,D.E.Evans,Y.Kawahigashi: "On α-induction,chiral generators and modular invariants for subfactors"Communications in Mathematical Physics. 208. 429-487 (1999)

J.Bockenhauer、D.E.Evans、Y.Kawahigashi：“关于 α 归纳、手性生成器和子因子的模不变量”数学物理通讯 208. 429-487 (1999)

D.E.Evans Y.Kawahigashi: "Quantum Symmotries on operator algebras"Oyford University Press. 848 (1998)

D.E.Evans Y.Kawahigashi：“算子代数的量子对称”奥伊福德大学出版社。

Y.Kawahigashi, R.Longo and M.Muger: "Multi-interval subfactors and modularity of representations in conformal field theory"Commun.Math.Phys.. (to appear).

Y.Kawahigashi、R.Longo 和 M.Muger：“共形场理论中的多区间子因子和表示模块性”Commun.Math.Phys..（待发表）。