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2001 Fiscal Year Final Research Report Summary

Study of integrable structure inquanteem integrable model

Research Project

Project/Area Number 10640016
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionNagoya University

Principal Investigator

NAKANISHI Tomoki  Graduate School of Mathematics,Associate Professor, 大学院・多元数理科学研究科, 助教授 (80227842)

Co-Investigator(Kenkyū-buntansha) OKADA Soichi  Graduate School of Mathematics,Associate Professor, 大学院・多元数理科学研究科, 助教授 (20224016)
TSUCHIYA Akihiro  Graduate School of Mathematics,Professor, 大学院・多元数理科学研究科, 教授 (90022673)
AOMOTO Kazuhiko  Graduate School of Mathematics,Professor, 大学院・多元数理科学研究科, 教授 (00011495)
HAYASHI Takahiro  Graduate School of Mathematics,Associate Professor, 大学院・多元数理科学研究科, 助教授 (60208618)
Project Period (FY) 1998 – 2001
KeywordsBethe ansatz / Heisenberg model / quantum group / intergrable models
Research Abstract

We obtain the following new results on the integrable structure of integrable lattice models.
1. The formal completeness theorem on the Bethe equation for the XXZ-type spin models. (Nakanishi, Kuniba, Tsuboi (Tokyo Univ.))
By classifying the string solutions of the Bethe equation for the XXZ-type spin models in the q=0 limit, we showed that, under the Kirillov-Reshetikhin (KR) conjecture on the KR modules of the quantum affine algebras, the power series formula of the character of the KR module representing the formal completeness of the XXZ-type spin models holds for any affine Lie algebra.
2. Reformulation of the Kirillov-Reshetikhin conjecture by the canonical solutions of the Q-systems. (Nakanishi, Kuniba, Tsuboi (Tokyo Univ.))
By observing the principle mechanism for various formulae and theorems obtained in the study of 1, we completely clarified the relation between these power series formulae representing the formal completeness and the underlying functional (algebraic) equations (Q-system of KR-type). Namely, we introduce a kind of functional equations (finite Q-system) for a finite number of functions of a finite number of variables. A finite Q-system has a remarkable property that its unique solution admits two power series formula by the Lagrange inversion formula for several variables. They are exactly a finite-variable analogue of the formal completeness formulae for the XXX-type and XXZ-type models. The formal completeness formula can be obtained as the projective limit of this formula.

  • Research Products

    (8 results)

All Other

All Publications (8 results)

  • [Publications] T.Nakanishi, A.Kuniba: "The Bethe equation at q=0, the Mobius inversion formula, and weight multiplicities : II The X_n case"Journal of Algebra. (印刷中).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Nakanishi, A.Kuniba, Z.Tsuboi: "The Bethe equation at q=0, the Mobius inversion formula, and weight multiplicities : III The X^<(r)>_n case"Letters in Mathematical Physics. (印刷中).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Nakanishi, A.Kuniba, Z.Tsuboi: "The canonical solutions of the Q-systems and the Kirillov-Reshetikhin conjecture"Communications in Mathematical Physics. (印刷中).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] K.Aomoto, K.Iguchi: "Wu's equations and quasi Hypergeometric functions"Communications in Mathematical Physics. 223. 475-507 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Nakanishi (with A.Kuniba): "The Bethe equation at q=0, the Mobius inversion formula, and weight multiplicities : II THE X_n case"Journal of Algebra. (in press).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T.Nakanishi(with A.Kuniba, Z.Ysuboi): "The Bethe equation at q=0, the Mobius inversion formula, and weight multiplicities : III THE X_n case"Letters in Mathematical Physics. (in press).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T.Nakanishi(with A.kuniba, Z.Tsuboi): "The canonical solution of the Q-systems and the Kirillou-Reshetikhin conjectare"Communications in Mathematical Physics. (in press).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K.Aomoto(with K.Iguchi): "Wu's equations and quasihypergeometra funations"Communications in Mathematical Physics. 223. 475-507 (2001)

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2003-09-17  

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