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Studies of Algebraic Analysis

Research Project

Project/Area Number 63460005
Research Category

Grant-in-Aid for General Scientific Research (B)

Allocation TypeSingle-year Grants
Research Field 解析学
Research InstitutionKyoto University

Principal Investigator

KASHIWARA Masaki  Kyoto University. Research Institute for Mathematical Sciences, Professor, 数理解析研究所, 教授 (60027381)

Co-Investigator(Kenkyū-buntansha) NAKANISHI Noboru  Kyoto University. Research Institute for Mathematical Sciences, Professor, 数理解析研究所, 教授 (30027362)
TAKASAKI Kanehisa  Kyoto University. Research Institute for Mathematical Sciences, Instructor, 数理解析研究所, 助手 (40171433)
SAITO Morihiko  Kyoto University. Research Institute for Mathematical Sciences, Instructor, 数理解析研究所, 助手 (10186968)
MIWA Tetsuji  Kyoto University. Research Institute for Mathematical Sciences, Associate Profes, 数理解析研究所, 助教授 (10027386)
KAWAI Takahiro  Kyoto University. Research Institute of Mathematical Sciences, Professor, 数理解析研究所, 教授 (20027379)
島川 和久  京都大学, 数理解析研究所, 助手 (70109081)
荒木 不二洋  京都大学, 数理解析研究所, 教授 (20027361)
斎藤 恭司  京都大学, 数理解析研究所, 教授 (20012445)
大沢 健夫  京都大学, 数理解析研究所, 助教授 (30115802)
島田 信夫  京都大学, 数理解析研究所, 教授 (70027358)
Project Period (FY) 1988 – 1990
Project Status Completed (Fiscal Year 1990)
Budget Amount *help
¥5,200,000 (Direct Cost: ¥5,200,000)
Fiscal Year 1990: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 1989: ¥2,100,000 (Direct Cost: ¥2,100,000)
Fiscal Year 1988: ¥2,100,000 (Direct Cost: ¥2,100,000)
KeywordsKac-Moody algebra / Quantum group / Solvable lattice model / R matrix / Chiral Potts model / Hodge theory / unnarmonic oscillator / quantum gravity / 量子重力 / 無限次元多様体 / 結晶基底 / 共形場の理論 / 混合ホッジ加群 / 代数解析学 / 無限次元グラスマン多様体 / 無限次元リー環 / 共変量子力学
Research Abstract

Kashiwara proved the generalized Kazhdan-Lustzig conjecture for Kac-Moody Lie algebras. He also proved the existence of special basis with certain nice properties at q=0 for representations of quantum groups. He named it crystal base and developed the theory. Miwa showed that crystal base is equivalent to the paths appearing in the computation of 1 point function of solvable lattice models, and, by using this fact, constructed the crystal base for the affine Lie algebras of type A. He also studied the representation theory of quantum when q is a root of unity and the related lattice models in the collaboration with Jimbo, Date, Miki. When q is a root of unity, quantum groups have a large center. The representations are classified by the values of these central elements. The R matrices exist only if they are chosen appropriately and Riimann surfaces appear form this condition. The corresponding lattice models, Which are the generalized chiral potts models, Were constructed. Morihiko Saito formulated the mixed Hodge modules as characteristic 0 analogue of the mixed Hodge theory of Deligne et al. Kawai studied the WKB method in connection with the unharmonic oscillators with Aoki and Takei, and obtained the canonical form of Schrodinger equation in the case of 1 or 2 simple turning points. Takasaki also studied the WKB method connecting Olver's method and the method of Jost solution in the scattering theory, and obtained multiple integral series for the Voros coefficients. Nakanishi studied the covariant canonical quantization of quantum gravity with Abe. They succeeded to obtain a consistent 1 order perturbation in Einstein gravitational constant, Without using a c-number background metric.

Report

(4 results)
  • 1990 Annual Research Report   Final Research Report Summary
  • 1989 Annual Research Report
  • 1988 Annual Research Report
  • Research Products

    (27 results)

All Other

All Publications (27 results)

  • [Publications] 柏原 正樹: "Crystallizing the qーanalogue of Universal Enveloping Algebras" Preprint,RIMSー728. (1990)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1990 Final Research Report Summary
  • [Publications] 河合 隆裕: "On the existence of holomorphic solutions of systems of linear differential equations of infinite order and with constant coefficients" Int.J.oF Math.1. 63-82 (1990)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1990 Final Research Report Summary
  • [Publications] 三輪 哲二: "Braid Group Representations Arising from the Generalized Chiral Potts Models" Preprint,RIMSー729. (1990)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1990 Final Research Report Summary
  • [Publications] 斎藤 盛彦: "On the Injectivity of Cycle Maps" Preprint,RIMSー721. (1990)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1990 Final Research Report Summary
  • [Publications] 中西 襄: "Interchanging the roles of spacetime and FaddeevーPopov ghost in quantum Einstein gravity II" Progr.Theoret.Phys.83. 1054-1064 (1990)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1990 Final Research Report Summary
  • [Publications] KASHIWARA, Masaki: "Crystallizing the q-analogue of Universal Enveloping Algebras" Preprint. RIMS-728. (1990)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1990 Final Research Report Summary
  • [Publications] KAWAI, Takahiro: "On the existence of holomorphic solutions of systems of linear differential equations of infinite order and with constant coefficients" Int. J. of Math.Vol. 1. 63-82 (1990)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1990 Final Research Report Summary
  • [Publications] MIWA, Tetsuji: "Braid Group Representations Arising from the Generalized Chiral Potts Models" Preprint. RIMS-729. (1990)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1990 Final Research Report Summary
  • [Publications] SAITO, Morihiro: "On the Injectivity of Cycle Map" Preprint. RIMS-721. (1990)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1990 Final Research Report Summary
  • [Publications] NAKANISHI, Noboru: "Interchanging the roles of spacetime and Faddeev-Popov ghost in quantums Einstein gravity II" Progr. Teoret. Phys.83. 1054-1064 (1990)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1990 Final Research Report Summary
  • [Publications] 柏原 正樹: "Crystallizing the qーanaloque of Universal Enveloping Algebras" Preprint,RIMSー728. (1990)

    • Related Report
      1990 Annual Research Report
  • [Publications] 河合 隆裕: "On the existence of holomorphic solutions of systems of linear differential equaitions of infinite order and with constant coefficients" Int.J.of Math.1. 63-82 (1990)

    • Related Report
      1990 Annual Research Report
  • [Publications] 三輪 哲二: "Braid Group Representations Arising from the Generalized Chiral Potts Models" Preprient,RIMSー729. (1990)

    • Related Report
      1990 Annual Research Report
  • [Publications] 斎藤 盛彦: "On the Injectivity of Cycle Maps" Preprint,RIMSー721. (1990)

    • Related Report
      1990 Annual Research Report
  • [Publications] 中西 襄: "Interchanging the roles of spacetime and FaddeevーPopov ghost in quantum Einstein gravity II" Progr.Theoret.Phys.83. 1054-1064 (1990)

    • Related Report
      1990 Annual Research Report
  • [Publications] 柏原正樹: "Crystalizing the q-analogue of universal enveloping algebra" MSP.

    • Related Report
      1989 Annual Research Report
  • [Publications] 荒木不二洋: "An application of Dye's theorem on projective lattice to orthogonally decomposable isomorphisms" Pac.J.Math.137. 1-13 (1981)

    • Related Report
      1989 Annual Research Report
  • [Publications] 中西襄: "Interchanging the roles of space-time and Faddeev-Popov ghost in quantum Einstein gravity" Prog.Theor.Phys.83. (1990)

    • Related Report
      1989 Annual Research Report
  • [Publications] 河合隆裕: "A particular partition of unity-an auxiliary tool in Hodge theory" Proc.Symp.in Pure Math.49. 19-26 (1989)

    • Related Report
      1989 Annual Research Report
  • [Publications] 高崎金久: "An infinite number of hidden variables in hyper-Kahler metrics" J.Math.Phys.30. 1515-1521 (1989)

    • Related Report
      1989 Annual Research Report
  • [Publications] 島川和久: "Infinite Loop G-spaces associated to monoidal G-graded category" Publ.RIMS. 25. 239-262 (1989)

    • Related Report
      1989 Annual Research Report
  • [Publications] Masaki,KASHIWARA: American Journal of Mathemetics.

    • Related Report
      1988 Annual Research Report
  • [Publications] Takahiro,KAWAI;H.Stapp: Algebraic Analysis. 1. 309-330 (1988)

    • Related Report
      1988 Annual Research Report
  • [Publications] Takeo,OHSAWA: Publication of RIMS,Kyoto Univ.24. 265-276 (1988)

    • Related Report
      1988 Annual Research Report
  • [Publications] Noboru,NAKANISHI;Mitsuo,ABE: Prog.Thoer.Phys.80. 913-921 (1988)

    • Related Report
      1988 Annual Research Report
  • [Publications] Kazuhisa,SHIMAKAWA: Publication of RIMS,Kyoto Univ.24. 99-120 (1988)

    • Related Report
      1988 Annual Research Report
  • [Publications] Kanahisa,TAKASAKI: Letters in Math.Physics.

    • Related Report
      1988 Annual Research Report

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Published: 1988-04-01   Modified: 2016-04-21  

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