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Integrable models and quantum cluster algebras

Research Project

Project/Area Number 24540203
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Global analysis
Research InstitutionThe University of Tokyo

Principal Investigator

KUNIBA ATSUO  東京大学, 総合文化研究科, 教授 (70211886)

Co-Investigator(Renkei-kenkyūsha) YAMADA Yasuhiko  神戸大学, 理学部, 教授 (00202383)
NAKANISHI Tomoki  名古屋大学, 多元数理研究科, 教授 (80227842)
OKADO Masato  大阪市立大学, 理学研究科, 教授 (70221843)
Project Period (FY) 2012-04-01 – 2015-03-31
Project Status Completed (Fiscal Year 2014)
Budget Amount *help
¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000)
Fiscal Year 2014: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2013: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2012: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Keywords可積分系 / 量子群 / ヤン・バクスター方程式 / 四面体方程式 / 量子座標環 / 3次元反射方程式 / PBW基底 / 表現論 / 量子アフィン・リー環 / q 振動子 / PBW 基底 / 3次元反射方程式 / 4面対方程式 / q-振動子表現 / 量子R行列 / 量子クラスター代数
Outline of Final Research Achievements

Several results illuminating novel mathematical structures of 3-dimensional integrable systems are obtained. (1) By invoking the representation theory of quantized coordinate ring of SP(4), two kinds of solutions are constructed for the 3-dimensional reflection equation proposed by Isaev and Kulish in 1997 for the first time. Its combinatorial limit, the classical analogue and a polynomial formula are obtained. (2) The transition coefficients of the PBW bases of the positive part of the quantized universal enveloping algebra Uq(g) are shown to coincide with the matrix elements of the intertwiner of the Soibelman representations of the quantized coordinate ring of g for all g of finite classical type. (3) For the two solutions known as R-operator and L-operator of the tetrahedron equation, 2-dimensional reduction is performed, and the results are identified with the quantum R matrices for generalized quantum groups including quantum superalgebras.

Report

(4 results)
  • 2014 Annual Research Report   Final Research Report ( PDF )
  • 2013 Research-status Report
  • 2012 Research-status Report
  • Research Products

    (32 results)

All 2015 2014 2013 2012 Other

All Journal Article (15 results) (of which Peer Reviewed: 13 results,  Acknowledgement Compliant: 2 results,  Open Access: 1 results) Presentation (16 results) (of which Invited: 12 results) Remarks (1 results)

  • [Journal Article] Tetrahedron equation and quantum R matrices for modular double of U_q(D^{(2)}_{n+1}), U_q(A^{(2)}_{2n}) and U_q(C^{(1)}_{n})2015

    • Author(s)
      Atsuo Kuniba, Masato Okado and Sergey Sergeev
    • Journal Title

      Letters in Mathematical Physics

      Volume: 105 Issue: 3 Pages: 447-461

    • DOI

      10.1007/s11005-015-0747-0

    • Related Report
      2014 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] A polynomial formula for the solution of 3D reflection equation2015

    • Author(s)
      Atsuo Kuniba and Shouya Maruyama
    • Journal Title

      Journal of Physics A: Mathematical and Theoretical

      Volume: 48 Issue: 13 Pages: 135204-135204

    • DOI

      10.1088/1751-8113/48/13/135204

    • Related Report
      2014 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Tetrahedron equation and quantum R matrices for q-oscillator representations2015

    • Author(s)
      Atsuo Kuniba and Masato Okado
    • Journal Title

      Journal of Physics: Conference Series

      Volume: 597 Pages: 012051-012051

    • DOI

      10.1088/1742-6596/597/1/012051

    • Related Report
      2014 Annual Research Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Journal Article] Cluster algebras in mathematical physics2014

    • Author(s)
      P. Di Francesco, M. Gekhtman, A. Kuniba and M. Yamazaki
    • Journal Title

      J. Phys. A: Math.Theor.

      Volume: 47 Issue: 47 Pages: 470301-470301

    • DOI

      10.1088/1751-8113/47/47/470301

    • Related Report
      2014 Annual Research Report
  • [Journal Article] Diagrammatic description of c-vectors and d-vectors of cluster algebras of finite type2014

    • Author(s)
      Nakanishi, T., Stella, S.
    • Journal Title

      The Electronic Journal of Combinatorics

      Volume: 21

    • Related Report
      2013 Research-status Report
    • Peer Reviewed
  • [Journal Article] Symmetries of quantum Lax equations for the Painlev\'e equations2014

    • Author(s)
      Nagoya, H., Yamada, Y.
    • Journal Title

      Annales Henri Poincare

      Volume: 15 Pages: 313-344

    • Related Report
      2013 Research-status Report
    • Peer Reviewed
  • [Journal Article] A common structure in PBW bases of the nilpotent subalgebra of U_q(g) and quantized algebra of functions2013

    • Author(s)
      A. Kuniba, M. Okado and Y. Yamada
    • Journal Title

      SIGMA

      Volume: 9

    • DOI

      10.3842/sigma.2013.049

    • Related Report
      2013 Research-status Report
    • Peer Reviewed
  • [Journal Article] A solution of the 3D reflection equation from quantized algebra of functions of type B2013

    • Author(s)
      Kuniba, A., Okado, M
    • Journal Title

      Proceedings of the XXIX International Colloquium on Group-Theoretical Methods in Physics

      Volume: 11

    • Related Report
      2013 Research-status Report
    • Peer Reviewed
  • [Journal Article] Tetrahedron equation and quantum R matrices for spin representations of B^{(1)}_n, D^{(1)}_n and D^{(2)}_{n+1}2013

    • Author(s)
      A. Kuniba and S. Sergeev
    • Journal Title

      Communications in Mathematical Physics

      Volume: 324 Issue: 3 Pages: 695-713

    • DOI

      10.1007/s00220-013-1808-9

    • Related Report
      2013 Research-status Report
    • Peer Reviewed
  • [Journal Article] Tetrahedron equation and quantum R matrices for infinite dimensional modules of U_q(A^{(1)}_1) and U_q(A^{(2)}_2)2013

    • Author(s)
      A. Kuniba and M. Okado
    • Journal Title

      Journal of Physics A: Mathematical and Theoretical

      Volume: 46 Issue: 48 Pages: 485203-485203

    • DOI

      10.1088/1751-8113/46/48/485203

    • Related Report
      2013 Research-status Report
    • Peer Reviewed
  • [Journal Article] Simplicity and similarity of Kirillov-Reshetikhin crystals2013

    • Author(s)
      Okado, M.
    • Journal Title

      Contemp. Math.

      Volume: 602 Pages: 183-194

    • Related Report
      2013 Research-status Report
    • Peer Reviewed
  • [Journal Article] Periodicities of T and Y-systems, dilogarithm identities, and cluster algebras I: Type B_r2013

    • Author(s)
      Rei Inoue 他
    • Journal Title

      Publ. RIMS

      Volume: 49 Issue: 1 Pages: 1-42

    • DOI

      10.4171/prims/95

    • Related Report
      2012 Research-status Report
    • Peer Reviewed
  • [Journal Article] Periodicities of T and Y-systems, dilogarithm identities, and cluster algebras II: Types C_r, F_4, and G_22013

    • Author(s)
      Rei Inoue 他
    • Journal Title

      Publ. RIMS

      Volume: 49 Issue: 1 Pages: 43-85

    • DOI

      10.4171/prims/96

    • Related Report
      2012 Research-status Report
    • Peer Reviewed
  • [Journal Article] Tetrahedron and 3D reflection equations from quantized algebra of functions2012

    • Author(s)
      A. Kuniba and M. Okado
    • Journal Title

      J. Phys. A: Mathematical and Theoretical

      Volume: 45 Issue: 46 Pages: 465206-465206

    • DOI

      10.1088/1751-8113/45/46/465206

    • Related Report
      2012 Research-status Report
    • Peer Reviewed
  • [Journal Article] 徒然なるまま厳密解2012

    • Author(s)
      国場敦夫
    • Journal Title

      数理科学「物理と数学における厳密解-自然現象の解明に向けて-」

      Volume: 11月号 Pages: 5-6

    • Related Report
      2012 Research-status Report
  • [Presentation] 四面体方程式とq-振動子表現の量子R行列2014

    • Author(s)
      国場敦夫(尾角正人と連名)
    • Organizer
      日本数学会
    • Place of Presentation
      学習院大学目白キャンパス, 豊島区, 東京都
    • Related Report
      2013 Research-status Report
  • [Presentation] Quantum Lax pairs for Painlev\'e systems and their solutions2014

    • Author(s)
      山田泰彦
    • Organizer
      学振二国間交流事業共同研究ロシアとの共同研究(RFBR)「ゲージ理論と弦理論の双対性による可積分性の統合と進展」
    • Place of Presentation
      KKRびわこホテル,大津市,滋賀県
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] Combinatorial and representation theoretical problems arising from solvable lattice models (I), (II)2014

    • Author(s)
      尾角正人
    • Organizer
      The 6th TIMS-OCAMI-WASEDA Joint International Workshop on Integrable Systems and Mathematical Physics
    • Place of Presentation
      National Taiwan University, Taipei, Taiwan
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] Towrad 3D integrability from quantum group2013

    • Author(s)
      国場敦夫
    • Organizer
      Scientific Meeting 2013, Institute of Publishing
    • Place of Presentation
      Institute of Physics, London, UK
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] Physical mathematics of Bethe ansatz2013

    • Author(s)
      国場敦夫
    • Organizer
      25th IUPAP International Conference on Statistical Physics
    • Place of Presentation
      Seoul National University, Seoul, Korea
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] Wonder of sine-Gordon Y-system2013

    • Author(s)
      中西知樹
    • Organizer
      Algebra, Combinatorics and Representation Theory: in honor of the 60th birthday of Andrei Zelevinsky,
    • Place of Presentation
      Northeastern University, Boston, USA
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] Diagrammatic description of c-vectors and d-vectors in cluster algebras2013

    • Author(s)
      中西知樹
    • Organizer
      String theory, integrable systems and representation theory
    • Place of Presentation
      京都大学数理解析研究所, 京都市, 京都府
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] U^+_q のPBW基底と量子座標環2013

    • Author(s)
      尾角正人(国場敦夫,山田泰彦と連名)
    • Organizer
      日本数学会
    • Place of Presentation
      愛媛大学城北キャンパス, 松山市, 愛媛県
    • Related Report
      2013 Research-status Report
  • [Presentation] Lax pairs for quantum Painlevé equations and their solutions2013

    • Author(s)
      山田泰彦
    • Organizer
      Workshop on Integrable Systems
    • Place of Presentation
      University of Sydney, Sydney, Australia
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] Box-ball system and soliton tau function2013

    • Author(s)
      山田泰彦
    • Organizer
      Around Sato's Theory on Soliton Equations
    • Place of Presentation
      津田塾大学, 小平市, 東京都
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] Tetrahedron equation and quantum groups2013

    • Author(s)
      尾角正人
    • Organizer
      Shanghai Workshop on Representation Theory
    • Place of Presentation
      East China Normal University, Shanghai, China
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] PBW bases and quantized algebra of functions2013

    • Author(s)
      尾角正人
    • Organizer
      String theory, integrable systems and representation theory
    • Place of Presentation
      京都大学数理解析研究所, 京都市, 京都府
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] 量子座標環と3次元反射方程式2013

    • Author(s)
      国場敦夫,尾角正人
    • Organizer
      日本数学会
    • Place of Presentation
      京都大学吉田キャンパス
    • Related Report
      2012 Research-status Report
  • [Presentation] Tetrahedron and 3D reflection equations from quantized algebra of functions2012

    • Author(s)
      A. Kuniba
    • Organizer
      The XXIX International Colloquium on Group-Theoretical Methods in Physics
    • Place of Presentation
      Chern Institute of Mathematics, Tianjin, China
    • Related Report
      2012 Research-status Report
    • Invited
  • [Presentation] Tetrahedron Equation and Quantum R Matrices for Spin Representations2012

    • Author(s)
      国場敦夫,Sergey Sergeev
    • Organizer
      日本数学会
    • Place of Presentation
      九州大学伊都キャンパス
    • Related Report
      2012 Research-status Report
  • [Presentation] Tetrahedron and 3D reflection equations from quantized algebra of functions2012

    • Author(s)
      国場敦夫
    • Organizer
      組合せ論的表現論とその周辺
    • Place of Presentation
      京大数理研
    • Related Report
      2012 Research-status Report
    • Invited
  • [Remarks] Main publications with capricious comments

    • URL

      http://maildbs.c.u-tokyo.ac.jp/~kuniba/atsuo/publications.html

    • Related Report
      2014 Annual Research Report

URL: 

Published: 2013-05-31   Modified: 2019-07-29  

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