Classificaition of subfactors in operator algebra and its applications
Project/Area Number  10640200 
Research Category 
GrantinAid for Scientific Research (C).

Section  一般 
Research Field 
Global analysis

Research Institution  University of Tokyo 
Principal Investigator 
KAWAHIGASHI Yasuyuki University of Tokyo Graduate School of Mathematical Sciences Professor, 大学院・数理科学研究科, 教授 (90214684)

Project Fiscal Year 
1998 – 2000

Project Status 
Completed(Fiscal Year 2000)

Budget Amount *help 
¥3,200,000 (Direct Cost : ¥3,200,000)
Fiscal Year 2000 : ¥800,000 (Direct Cost : ¥800,000)
Fiscal Year 1999 : ¥1,100,000 (Direct Cost : ¥1,100,000)
Fiscal Year 1998 : ¥1,300,000 (Direct Cost : ¥1,300,000)

Keywords  operator algebra / subfactor / conformal field theory / alphainibuction / quantum double / 作用素環 / Subfactor / alphainduction / Longo / induction / modular invariant / conformal field / Gaiois correspondence / paragroup / 部分因子環 / Dynkin 図形 / conformal inclusion / loop group 
Research Abstract 
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 LongoRehren. Bockenhauer, Evans and I have extended definitions of both the αinduction 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 LongoRehren construction and prove that nondegeneracy of a braiding holds automatically in this setting. We have determined the structure of MM fusion rule algebras arising from chiral αinduction using one braiding in terms of chiral branching coefficients. As applications, we have determined the full MM 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 LongoRehren subfactors arising from αinduction. We can describe the tensor categories arising from the LongoRehren subfactors. We have further shown that if the braiding is nondegenerate, then the subfactor we obtain as a dual of the usual LongoRehren subfactor after αinduction is isomorphic to the one arising from the generalized LongoRehren construction.

Report
(5results)
Research Output
(21results)