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Structure and symmetries of integrable systems

Research Project

Project/Area Number 05402001
Research Category

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

Allocation TypeSingle-year Grants
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionKYOTO UNIVERSITY

Principal Investigator

JIMBO Michio  Kyoto Univ.Graduate School of Science Professor, 大学院理学研究科, 教授 (80109082)

Co-Investigator(Kenkyū-buntansha) KONO Akira  Kyoto Univ., Graduate School of Science Professor, 大学院理学研究科, 助教授 (00093237)
UENO Kenji  Kyoto Univ., Graduate School of Science Professor, 大学院理学研究科, 助教授 (40011655)
UMEDA Toru  Kyoto Univ., Graduate School of Science Associate Professor, 大学院理学研究科, 助教授 (00176728)
TAKEI Yoshitsugu  Kyoto Univ., Research Institute for Mathematical Science Associate Professor, 数理解析研究科, 助教授 (00212019)
SHIOTA Takahiro  Kyoto Univ., Graduate School of Science Associate Professor, 大学院理学研究科, 助教授 (20243008)
斉藤 政彦  京都大学, 大学院・理学研究科, 助教授 (80183044)
野村 隆昭  京都大学, 理学部, 助教授 (30135511)
松沢 淳一  京都大学, 理学部, 助手 (00212217)
Project Period (FY) 1993 – 1995
Project Status Completed (Fiscal Year 1995)
Budget Amount *help
¥12,700,000 (Direct Cost: ¥12,700,000)
Fiscal Year 1995: ¥3,600,000 (Direct Cost: ¥3,600,000)
Fiscal Year 1994: ¥3,200,000 (Direct Cost: ¥3,200,000)
Fiscal Year 1993: ¥5,900,000 (Direct Cost: ¥5,900,000)
Keywordslattice model / correlations / quantum group / matrix integral / Painleve equation / q-difference operator / moduli space / loop group / qKZ方程式 / Capelli恒等式 / ゲージ群 / アーベル多様体 / 既約ユニタリ表現 / 可解格子模型 / 頂点作用素 / KP階層 / 無限次元代数 / 不変式 / Bott周期性 / 楕円型量子群 / Jordan代数 / dual pair / アフィン・ルート系 / ループ空間
Research Abstract

The major outcome of the present research project is as follows.
1.Jimbo pushed forward the study of space of states of lattice models. Ising and RSOS type models were formulated in much the same way as for the vertex type models, and difference equations for correlations were derived. The theory was extended to spin chains with a boundary, and physical quantities were derived including vacuum states, energy and magnetization. For critical systems an integral formula for correlations was found. In representation theory, a new level 0 action was constructed on a level one module over the quantum affine algebra.
2.Shiota constructed a notrivial solution to [P,Q] =P using a matrix integral similar to Kontsevich's. Takei showed, usuing the exact WKB analysis, that the Painleve I equation can be taken as a standard form near a simple turning point. He also constructed a general solution using the multiple scale anaysis.
3.Umeda studied a new construction of a q-analog of differential operators, which appear in the Capelli identity for GLq (n) , in terms of classical q-difference operators. This opened up a connection with Gelfand-type hypergeometric equations.
4.Ueno gave a concrete construction of projective flat connections on the vector bundles of conformal blocks over the moduli space of curves. Kono studied the free loop group of a compact simple Lie group, and found a connection between the mod p cohomology of the adjoint action of G on the closed loop group, and the integral cohomology of G.

Report

(4 results)
  • 1995 Annual Research Report   Final Research Report Summary
  • 1994 Annual Research Report
  • 1993 Annual Research Report
  • Research Products

    (37 results)

All Other

All Publications (37 results)

  • [Publications] M. Jimbo et al.: "Difference equations in spin chains with a boundary" Nucl. Phys.B448. 429-456 (1995)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1995 Final Research Report Summary
  • [Publications] M. Adler et al.: "A matrix integral solution to[P.Q]=P and matrix Laplace transforms" to appear in Commun. Math. Phys.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1995 Final Research Report Summary
  • [Publications] Y. Takei: "On exact WKB analysis for Painleve equations" Progr. Theoret. Phys. Suppl.116. 311-317 (1994)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1995 Final Research Report Summary
  • [Publications] M. Noumi et al.: "A quantum analogue of the Capelli identity and an elementary differential calculus on GLg(n)" Duke Math. J.76. 567-594 (1994)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1995 Final Research Report Summary
  • [Publications] K. Ueno: "Q-suructure of conformal field theory" Progress in Math. 129. 511-531 (1995)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1995 Final Research Report Summary
  • [Publications] A. Kono et al: "The adjoint action of a Lie group on the space of loops" J. Math. Soc. Japan. 45. 495-510 (1993)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1995 Final Research Report Summary
  • [Publications] M. Jimbo et al: "Algebraic Analysis of Solvable attice Models" American Mathematical Society, 152 (1994)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1995 Final Research Report Summary
  • [Publications] 上野健爾: "代数幾何入門" 岩波書店, 342 (1995)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1995 Final Research Report Summary
  • [Publications] M.Jimbo et al.: "Difference Equations in Spin Chains with a Boundary" Nucl.Phys.B448 [FS]. 429-456 (1995)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1995 Final Research Report Summary
  • [Publications] M.Adler et al.: "A matrix integral solution to [P,Q] =P and matrix Laplace transforms" Commun.Math.Phys.(To appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1995 Final Research Report Summary
  • [Publications] Y.Takei: "On exact WKB analysis for Painleve equations" Progr.Theoret.Phys.Suppl., 116. 311-317 (1994)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1995 Final Research Report Summary
  • [Publications] M.Noumi et al.: "A quantum analogue of the Capelli identity and an elementary differential calculus on GL_q (n)" Duke Math.J.76. 567-594 (1994)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1995 Final Research Report Summary
  • [Publications] K.Ueno: "Q-structure of comformal field theory" Progress in Math.129. 511-531 (1995)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1995 Final Research Report Summary
  • [Publications] A.Kono et al.: "The adjoint action of a Lie group on the space of loops" J.Math.Soc.Japan. 45. 495-510 (1993)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1995 Final Research Report Summary
  • [Publications] M.Jimbo et al.: "Algebraic Analysis of Solvable Lattice Models" CBMS Regional Conference Series in Mathematics. 85. AMS (1994)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1995 Final Research Report Summary
  • [Publications] K.Ueno: Introduction to Algebraic Geometry. Iwanami, (1995)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1995 Final Research Report Summary
  • [Publications] M. Jimbo et al.: "A q-analog of the sixth Painleve equation" to appear in Lett. Math. Phys.

    • Related Report
      1995 Annual Research Report
  • [Publications] M. Adler et al.: "A marix integral solution to [P,Q]=P and matrix Laplace transform" to appear in Commun. Math. Phys.

    • Related Report
      1995 Annual Research Report
  • [Publications] T. Aoki et al.: "WKB analysis of Painleve transcendents with a large parameter II" to appear in “Structure of solutions of differential equations", world scientific.

    • Related Report
      1995 Annual Research Report
  • [Publications] M. Noumi et al.: "A quantum dual pair(sl_2, on) and the associated Capelli Identity" Lett. Math. Phys.34. 1-8 (1995)

    • Related Report
      1995 Annual Research Report
  • [Publications] A. Kono et al.: "A remark on the homotopy type of certain gauge groups" to appear in J. Math. Kyoto Univ.

    • Related Report
      1995 Annual Research Report
  • [Publications] M. Saito: "On ipperbounds of Mordell-Weil ranks of higher genus fibrations" in “Algebraic cycles and related topics", World Scientific. 37-42 (1995)

    • Related Report
      1995 Annual Research Report
  • [Publications] 上野健爾: "代数幾何入門" 岩波書店, 342 (1995)

    • Related Report
      1995 Annual Research Report
  • [Publications] O.Foda et al.: "An elliptic quantum algebra for sl_2" Lett.Math.Phys.32. 259-268 (1994)

    • Related Report
      1994 Annual Research Report
  • [Publications] M.Jimbo et al.: "XXZ chain with a boundary" Nucl.Phys.to appear.

    • Related Report
      1994 Annual Research Report
  • [Publications] T.Shiota: "Calogero-Moser hierarchy and KP hierarchy" J.Math.Phys.35. 5844-5849 (1994)

    • Related Report
      1994 Annual Research Report
  • [Publications] M.Adler et al.: "From the Wes-algebra to its ceniral extension:a τ-function approach" Phys.Lett.A,to appear.

    • Related Report
      1994 Annual Research Report
  • [Publications] T.Nomura: "Grassmann inanifold of a JH-algebra" Ann.Global Anal.Geom.12. 237-260 (1994)

    • Related Report
      1994 Annual Research Report
  • [Publications] A.Kono et al.: "A topological proof of Bott periodicity theorem and a characterization of IBU" J.Math.Kyoto Univ. 34. 873-880 (1994)

    • Related Report
      1994 Annual Research Report
  • [Publications] M.Jimbo: "Algebraic Analysis of Solvable Lathice Models" CBMS regional conference series in Math,American Math.Society, 152 (1994)

    • Related Report
      1994 Annual Research Report
  • [Publications] M.Jimbo,T.Kojima,T.Miwa & Y.H.Quano: "Smirnov's integrals and quantum Knizhnik-Zamolodchikov equation of level 0" to appear in J.Phys.A.

    • Related Report
      1993 Annual Research Report
  • [Publications] T.Nomura: "Manifold of primitive idempotents in a Jordan-Hilbert algebra" J.Math.Soc.Japan. 45. 37-58 (1993)

    • Related Report
      1993 Annual Research Report
  • [Publications] M.Noumi,T.Umeda & M.Wakayama: "A quantum dual pair(sl_2,o_n)and the associated Capelli identity" to appear in Lett.Math.Phys.

    • Related Report
      1993 Annual Research Report
  • [Publications] J.Matsuzawa: "Root systems and periods on Hirzebruch surfaces" Publ.RIMS. 29. 411-438 (1993)

    • Related Report
      1993 Annual Research Report
  • [Publications] A.Kono & K.Kozima: "The adjoint action of a Lie group on the space of loops" J.Math.Soc.Jpa. 45. 495-510 (1993)

    • Related Report
      1993 Annual Research Report
  • [Publications] K.Inoue & A.Kono: "Nilpotency of a kernel of the Quillen map" J.Math.Kyoto Univ.33. 1047-1055 (1993)

    • Related Report
      1993 Annual Research Report
  • [Publications] 三輪哲二,神保道夫,伊達悦朗: "ソリトンの数理(岩波応用数学講座3)" 岩波書店, 112 (1993)

    • Related Report
      1993 Annual Research Report

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Published: 1993-04-01   Modified: 2016-04-21  

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