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Discretization and quantization of integrable and isomonodromic systems

Research Project

Project/Area Number 17540185
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Global analysis
Research InstitutionTohoku University

Principal Investigator

KUROKI Gen  Tohoku University, Graduate School of Science, Research Assistant, 大学院理学研究科, 助手 (10234593)

Co-Investigator(Kenkyū-buntansha) HASEGAWA Koji  Tohoku University, Graduate School of Science, Lecturer, 大学院理学研究科, 講師 (30208483)
KIKUCHI Tetsuya  Tokyo University, Graduate School of Mathematical Sciences, COE fellow, 大学院数理科学研究科, 研究拠点形成特任研究員 (00374900)
Project Period (FY) 2005 – 2006
Project Status Completed (Fiscal Year 2006)
Budget Amount *help
¥2,600,000 (Direct Cost: ¥2,600,000)
Fiscal Year 2006: ¥1,200,000 (Direct Cost: ¥1,200,000)
Fiscal Year 2005: ¥1,400,000 (Direct Cost: ¥1,400,000)
Keywordsintegrable systems / isomonodromic systems / Painleve systems / discretization / quantization / conformal field theory / quantum group / パンルヴェ方程式 / 超離散化 / 表現論
Research Abstract

One of the aims of this research is to quantize discrete classical dynamical systems arinsing from Berenstein-Kazhdan geometric crystals. Kajiwara, Noumi, and Yamada (2001) constructed, for any positive integers m and n, the birational action of the direct product of the extended affine Weyl groups of A-type on the space of the (m, n)-matrices, which is an important example of the discrete classical dynamical system arising from a geometric crystal. Using the affine quantum groups of A-type, Kuroki has constructed, for mutually prime m and n, the quantization of the birational action. This result would be the first step for understanding the relationship between quantum groups and geometric crystals.
Furthermore, as a byproduct, he clarified the relationship between quantum groups and q-difference birational Weyl group actions (q-difference Painleve systems). He has shown that, for any symmetrizable generalized Cartan matrix (GCM), the q-difference quantum birational Weyl group action i … More s generated by the complex powers of the lower Chevalley generators in the quantum universal enveloping algebra and this construction reproduces the q-difference quantum birational actions constructed by Hasegawa. Thus we can understand q-difference quantum Painleve systems in the language of quantum groups.
He also has pointed out the importance of the quantum L-operators or quantum groups characterized by the ALBL=LCLD relations. By the FRT construction, quantum groups can be characterized by the RLL=LLR relations. We need, however, the more general ALBL=LCLD relations to deal with quantum systems with birational Weyl group actions. He conjectured that quantum invariant polynomials of the q-difference quantum birational Weyl group action are generated by the mutually commuting transfer matrices arising from a certain ALBL=LCLD relation.
He announced most of the results mentioned above in the international workshop "Exploration of New Structures and Natural Constructions in Mathematical Physics" at Nagoya University, 5-8 March 2007
Hasegawa (in his preprint 2007) has constructed, for any symmetrizable GCM, a q-difference quantum birational Weyl group action on the algebra characterized by truncated q-Serre relations and has quantized the Panleve VI system.
Kikuchi has shown that ordinary differential Painlve VI sysmte and the q-difference Painleve VI system can be identified with the similarity reductions of certain differential and q-difference soliton systems respectively. Less

Report

(3 results)
  • 2006 Annual Research Report   Final Research Report Summary
  • 2005 Annual Research Report
  • Research Products

    (9 results)

All 2007 2006 2005

All Journal Article (9 results)

  • [Journal Article] The sixth Painleve equation as similarity reduction of affine gl_3 Drinfeld-Sokolov hierarchy2007

    • Author(s)
      Kakei, S., Kikuchi, T.
    • Journal Title

      Lett. Math. Phys. 79・3

      Pages: 221-234

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Annual Research Report 2006 Final Research Report Summary
  • [Journal Article] The sixth Painleve equation as similarity reduction of affine gl_3 Drinfeld-Sokolov hierarchy2007

    • Author(s)
      Kakei, S., Kikuchi, T.
    • Journal Title

      Lett. Math. Phys. Vol. 79, No. 3

      Pages: 221-234

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] A q-analogue of affine gl_3 hierarchy, and q-Painleve VI2006

    • Author(s)
      Kakei, S., Kikuchi, T.
    • Journal Title

      J. Phys. A : Math. Gen. 39

      Pages: 12179-12190

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] A q-analogue of affine gl_3 hierarchy and q-Painleve VI2006

    • Author(s)
      Kakei, S., Kikuchi, T.
    • Journal Title

      J. Phys. A : Math. Gen. Vol. 39

      Pages: 12179-12190

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] A q-analogue of affine gl_3 hierarchy and q-Painleve VI2006

    • Author(s)
      Kakei, S., Kikuchi, T.
    • Journal Title

      J. Phys. A : Math. Gen. 39

      Pages: 12179-12190

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Solutions of a derivative nonlinear Schroedinger hierarchy and its similarity reduction2005

    • Author(s)
      Kakei, S., Kikuchi, T.
    • Journal Title

      Glasg. Math. J. 47・A

      Pages: 99-107

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Solutions of a derivative nonlinear Schroedinger hierarchy and its similarity reduction2005

    • Author(s)
      Kakei, S., Kikuchi, T.
    • Journal Title

      Glasg. Math. J. Vol. 47, No. A

      Pages: 99-107

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Solution of a derivative nonlinear Schrodinger hierarchy and its similarity reduction2005

    • Author(s)
      S.Kakei, T.Kikuchi
    • Journal Title

      Glasgow Math.J. 47A

      Pages: 99-107

    • Related Report
      2005 Annual Research Report
  • [Journal Article] 一般的ドリンフェルト・ソコロフ階層の離散化と相似簡約2005

    • Author(s)
      算三郎, 菊地哲也
    • Journal Title

      数理解析研究所講究録 1422

      Pages: 192-203

    • Related Report
      2005 Annual Research Report

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Published: 2005-04-01   Modified: 2016-04-21  

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