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Classificaition of subfactors in operator algebra and its applications

Research Project

Project/Area Number 10640200
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Global analysis
Research InstitutionUniversity 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 Period (FY) 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)
Keywordsoperator 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
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 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.

Report

(4 results)
  • 2000 Annual Research Report   Final Research Report Summary
  • 1999 Annual Research Report
  • 1998 Annual Research Report
  • Research Products

    (21 results)

All Other

All Publications (21 results)

  • [Publications] D.E.Evans Y.Kawahigashi: "Orbifold Subfactors from Hecke algebvas II"Commun.Math.Phys.. 196. 331-361 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Y.Kawahigashi: "Quantum Galois cowespondance for subfactors"J.Funct.Anal.. 167. 481-497 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] J.Bockanlaner D.E.Evans.Y.Kawahigashi: "On α-induction, chiral projectors and modular invariants for subfactors"Commun.Math.Phys.. 208. 429-487 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Y.Kawalingashi,R.Longo M.,Mciger: "Multi-intervol subtracters and modularity of representations in carformol field theory"Commun.Math.Phys. (印刷中).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] J.BSckarhaner D.E.Evans C.Kawahigashi: "Chirol Structure of modular invaviants for subfactors"Commun.Math.Phys.. 210. 733-784 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] J.Bockonhaner D.E.Evaus.Y.Kawahigashi: "Lorgo-Rehvar subfactors arising from α-induction"Publ.RIMS Kyoto Univ.. (印刷中).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] D.E.Evans Y.Kawahigashi: "Quantum Symmotries on operator algebras"Oyford University Press. 848 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] D.E.Evans and Y.Kawahigashi: "Orbifold subfactors from Hecke algebras II"Commun.Math.Phys.. 196. 331-361 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Y.Kawahigashi: "Quantum Galois correspondence for subfactors"J.Funct.Anal.. 167. 481-497 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] J.Bockenhauer, D.E.Evans and Y.Kawahigashi: "On α-induction, chiral projectors, double triangle algebras and modular invariants for subfactors."Commun.Math.Phys.. 208. 429-487 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Y.Kawahigashi, R.Longo and M.Muger: "Multi-interval subfactors and modularity of representations in conformal field theory"Commun.Math.Phys.. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] J.Bockenhauer, D.E.Evans and Y.Kawahigashi: "Chiral structure of modular invariants for subfactors"Commun.Math.Phys.. 210. 733-784 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] J.Bockenhauer, D.E.Evans and Y.Kawahigashi: "Longo-Rehren subfactors arising from α-induction"Publ.RIMS, Kyoto Univ.. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] D.E.Evans and Y,Kawahigashi: "Quantum symmetries on operator algebras"Oxford University Press. 848 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] J.Bockenhauer,D.E.Evans,Y.Kawahigashi: "Longo-Rehren subfactors arising from α-induction"Publ,RIMS,Kyoto Univ.. (印刷中).

    • Related Report
      2000 Annual Research Report
  • [Publications] Y. Kawahigashi: "Quantum Galois currespondence for subfactors"Journal of Functional Analysis. 167. 481-497 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] 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)

    • Related Report
      1999 Annual Research Report
  • [Publications] J.Bockenhauer,D.E.Evans,Y.Kawahigashi: "Chiral structure of modular invariants for subfactors"Communications in Mathematical Physics. (印刷中).

    • Related Report
      1999 Annual Research Report
  • [Publications] D.E.Evans and Y.Kawahigashi: "Orbifold subfactor from Hecke algebras II" Communications in Mathematical Physics. 196. 331-361 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] Y.Kawahigashi: "Subfactor and paragrowp theory" Contenporary Mathematics. 228. 179-188 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] D.E.Evans and Y.Kawahigashi: "Quantum symmetries on operator algebras" Oxford University Press, 829 (1998)

    • Related Report
      1998 Annual Research Report

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Published: 1998-04-01   Modified: 2021-06-08  

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