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Algebraic analysis of infinite symmetry

Research Project

Project/Area Number 08454006
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionKYOTO UNIVERSITY

Principal Investigator

KASHIWARA Masaki  Professor, RIMS,Kyoto U., 数理解析研究所, 教授 (60027381)

Co-Investigator(Kenkyū-buntansha) IHARA Yasutaka  Professor, RIMS,Kyoto U., 数理解析研究所, 教授 (70011484)
MORI Shigefumi  Professor, RIMS,Kyoto U., 数理解析研究所, 教授 (00093328)
SAITO Kyoji  Professor, RIMS,Kyoto U., 数理解析研究所, 教授 (20012445)
DING Jintai  Lecturer, RIMS,Kyoto U., 数理解析研究所, 講師 (50273529)
MIWA Tetsuji  Professor, RIMS,Kyoto U., 数理解析研究所, 教授 (10027386)
Project Period (FY) 1996
Project Status Completed (Fiscal Year 1996)
Budget Amount *help
¥5,400,000 (Direct Cost: ¥5,400,000)
Fiscal Year 1996: ¥5,400,000 (Direct Cost: ¥5,400,000)
Keywordsalgebraic analysis / infinite symmetry / solvable model / quantum group / crystal base / correlation function
Research Abstract

1. Kazhdan-Lusztig Conjecture Lusztig conjectured that the irreducible character of highest weight modules with negative level is given by the Kazhdan-Lusztig polynomials. This conjecture had been proved in the integral level case. We succeeded to prove this conjecture in the non-integral case. This leads us to prove another conjecture of Lusztig on the modular representation of Chevalley groups.
2. Study of Crystal bases We succeeded to construct the Fock representations of the quantum affine algebras starting from an arbitrary finite-dimensional representations with perfect crystal. This is obtained as an analytic continuation from |q|<< 1. Furthermore we showed that this Fock representation decomposes to the tensor product of the Boson Fock space and the irreducible highest modules of the quantum affine algebras.
These results are recently applied to the study of modular representations of spin symmetric groups by the young French mathematicians Leclerc-Thibon.
3. Study of solvable models The solution to the q-KZ equation has been known when its parameter q satisfies |q|< 1. We constructed its solution when |q|=1 and studied its properties. This solution is supposed to be the correlation function of XXZ-model in the gap-less regime and we expect its further development.
4. Study of vertex operators Vertex operator is an operator on the irreducible highest weight modules indexed by finite-dimensional representation of quantum affine algebra. We analyze the vertex operator via crystal bases and leads to a new class of representations of quantum affine algebras.

Report

(2 results)
  • 1996 Annual Research Report   Final Research Report Summary
  • Research Products

    (20 results)

All Other

All Publications (20 results)

  • [Publications] Kashiwara, Masaki and Schapira, Pierre: "Moderate and formal cohomology associted with constructible sheaves" Mem.Soc.Math.France. 64. (1996)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] Kashiwara, Masaki and Tanisaki, Toshiyuki: "Kazhdan-Lusztig conjecture for affine Lie algebra with negative level,II,Non-integral case" Duke Math.J.84. 771-813 (1996)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] Jimbo, Michio and Miwa, Tetsuji: "Quantum KZ equation with |q|=1 and correlation function of the XXZ model in the gapless regime" J.Phys.A:Math.Gen.29. 2923-2958 (1996)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] Ihara, Yasutaka and Matsumoto, Makoto: "On Galois actions on profinite completions of braid groups" Contemp.Math.186. 173-200 (1995)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] Haag, Rudolf and Ojima, Izumi: "On the Problem of Defining a Specific Theory within the Frame of Local Quantum Physics." Ann.Inst.Henri Poincare. 64, 4. 385-393 (1996)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] Iwata, Satoru and Murota, Kazuo: "Horizontal principal structure of layered mixed.matrices : Decomposition of discrete systems by design-variable selections" SIAM J.Sci. Discrete Math.9, 1. 71-86 (1996)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] 大矢 雅則: "量子情報と進化の力学" 牧野書店, 270 (1996)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] Kashiwara, Masaki and Schapira, Pierre: "Moderate and formal cohomology associated with constructible sheaves" Mem.Soc.Math.France. 64. (1994)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] Kashiwara, Masaki and Tanisaki, Toshiyuki: "Kazhdan-Lusztig conjecture for affine Lie algebra with negative level, II,Non-integral case" Duke Math.J.84. 771-813 (1996)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] Jimbo, Michio and Miwa, Tetsuji: "Quantum KZ equation with |q|=1 and correlation function of the XXZ model in the gapless regime" J.Phys.A : Math.Gen.29. 2923-2958 (1996)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] Ihara, Yasutaka and Matsumoto, Makoto: "On Galois actions on profinite completions of braid groups" Contemp.Math.186. 173-200 (1995)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] Haag, Rudolf and Ojima, Izumi: "On the Problem of Defining a Specific Theory within the Frame of Local Quantum Physics" Ann.Inst.Henri Poincare. 64,4. 385-393 (1996)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] Iwata, Satoru and Murota, Kazuo: "Horizontal principal structure of layred mixed matrices : Decomposition of discrete systems by design-variable selections" SIAM J.Sci.Discrete Math.9,1. 71-86 (1996)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] Masaki Kashiwara: "Moderate and formal cohomology of Q associated with constructible sheaves" Memoire de la Soc.Math.de France. No.64. (1996)

    • Related Report
      1996 Annual Research Report
  • [Publications] Masaki Kashiwara: "Kazhdan-Lusztig cojecture for affine Lie algebra with negative level" Duke Math.J.Vol.84. 771-813 (1996)

    • Related Report
      1996 Annual Research Report
  • [Publications] Michio Jimbo: "Quantum KZ equation with |q|=1 and correlation function of the XXZ model in the gapless regime" J.Phys.A : Math.Gen.29. 2923-2958 (1996)

    • Related Report
      1996 Annual Research Report
  • [Publications] Yasutaka Ihara: "On Galois actions on profinite completions of braid groups" Contemp.Math. 186. 173-200 (1995)

    • Related Report
      1996 Annual Research Report
  • [Publications] Rudolf Haag: "On the Problem of Definig a Specific Theory within the Frame of Local Quantum Physics" Ann.Inst.Henri Poincare. 64.4. 385-393 (1996)

    • Related Report
      1996 Annual Research Report
  • [Publications] Satoru Iwata: "Horizontal principal structure of layered mixed matrices" SIAM Journal on Scientific Computing. 9.1. 71-86 (1996)

    • Related Report
      1996 Annual Research Report
  • [Publications] 大矢 雅則: "量子情報と進化の力学" 牧野書店, 270 (1996)

    • Related Report
      1996 Annual Research Report

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Published: 1996-04-01   Modified: 2016-04-21  

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