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Gepmrtic study of quantum groups and its applications

Research Project

Project/Area Number 16K05055
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Algebra
Research InstitutionRikkyo University (2017-2019)
The University of Tokyo (2016)

Principal Investigator

SAITO Yoshihisa  立教大学, 理学部, 教授 (20294522)

Project Period (FY) 2016-04-01 – 2020-03-31
Project Status Completed (Fiscal Year 2019)
Budget Amount *help
¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000)
Fiscal Year 2019: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2018: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2017: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2016: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Keywords量子包絡代数 / 量子包絡環 / 結晶基底 / 量子群
Outline of Final Research Achievements

In resent study of mathematical physics (string theory, especially), it is known that toroidal Lie algebras (LTA), and quantum toroidal algebras (QTA) play important roles. In this reseach project, we investigate the structure of these algebras in uniform way, by using the theory of elliptic root systems. Especially, we prove that these algebras have an elliptic modular group action induced from one on elliptic root systems. We believe these properties are quite
important in the future study of these algebras.

Academic Significance and Societal Importance of the Research Achievements

本研究の対象となるTLA及びQTAは,近年弦理論を始めとして様々な分野で注目を浴びている代数系であるが,既存の手法が殆ど役に立たないとの理由から組織的な研究は殆ど行われて来なかった.本研究では楕円ルート系の理論を用いてこの代数系を調べ,楕円モジュラー群がTLA, QTAに作用することを示した.これは,既存のルート系の理論,リー代数,量子包絡代数には無かった全く新しい性質であり,今後の当該代数の研究に重要な役割を果たすことが期待される.

Report

(5 results)
  • 2019 Annual Research Report   Final Research Report ( PDF )
  • 2018 Research-status Report
  • 2017 Research-status Report
  • 2016 Research-status Report
  • Research Products

    (17 results)

All 2020 2019 2018 2017 2016 Other

All Int'l Joint Research (3 results) Journal Article (3 results) (of which Int'l Joint Research: 1 results,  Peer Reviewed: 2 results,  Open Access: 2 results,  Acknowledgement Compliant: 1 results) Presentation (10 results) (of which Int'l Joint Research: 7 results,  Invited: 8 results) Remarks (1 results)

  • [Int'l Joint Research] Universite Lion 1(フランス)

    • Related Report
      2019 Annual Research Report
  • [Int'l Joint Research] University of Cologne(ドイツ)

    • Related Report
      2019 Annual Research Report
  • [Int'l Joint Research] Universite Lyon 1(フランス)

    • Related Report
      2018 Research-status Report
  • [Journal Article] Invariants of the Weyl group of type A_{2l}^{(2)}2020

    • Author(s)
      Kenji Iohara and Yoshihisa Saito
    • Journal Title

      Pure and Applied Mathematics Quarterly

      Volume: -

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Remarks on the Drinfeld realization of quantum affine algebras2019

    • Author(s)
      斉藤義久
    • Journal Title

      Algebraic Lie Theory and Representation Theory

      Volume: 4 Pages: 139-146

    • Related Report
      2018 Research-status Report
  • [Journal Article] Quantized coordinate rings, PBW-type bases and q-boson algebras2016

    • Author(s)
      Yoshihisa Saito
    • Journal Title

      Journal of Algebra

      Volume: 453 Issue: 1 Pages: 456-491

    • DOI

      10.1016/j.jalgebra.2011.01.010

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Presentation] 楕円Artin群について2020

    • Author(s)
      Yoshihisa Saito
    • Organizer
      2020年度日本数学会年会,無限可積分系セッション特別公演
    • Related Report
      2019 Annual Research Report
    • Invited
  • [Presentation] Elliptic Artin groups2019

    • Author(s)
      Yoshihisa Saito
    • Organizer
      Geometry and Integrable Systems
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Elliptic Artin groups2019

    • Author(s)
      Yoshihisa Saito
    • Organizer
      Algebra und Zahlentheorie Seminare
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Artin groups associated to elliptic root systems2019

    • Author(s)
      Yoshihisa Saito
    • Organizer
      Conference on Algebraic Representation Theory 2019
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Remarks on the Drinfeld realization of quantum affine algebras2018

    • Author(s)
      Yoshihisa Saito
    • Organizer
      Algebraic Lie Theory and Representation Theory 2018
    • Related Report
      2018 Research-status Report
  • [Presentation] On quantum toroidal algebras associated with arbitrary root systems2018

    • Author(s)
      Yoshihisa Saito
    • Organizer
      Arbeitsgruppe Algebra und Zahlentheorie Seminare, Mathematisches Institut, University of Cologue.
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] On quantum toroidal algebras associated with arbitrary semisimple Lie algebras2017

    • Author(s)
      Yoshihisa Saito
    • Organizer
      Infinite Analysis 2017
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] On quantum elliptic algebras2017

    • Author(s)
      Yoshihisa Saito
    • Organizer
      MS Seminar
    • Place of Presentation
      Kavli IPMU, University of Tokyo
    • Related Report
      2016 Research-status Report
  • [Presentation] Quantum groups, quivers and related geometry2016

    • Author(s)
      Yoshihisa Saito
    • Organizer
      Berkeley-Tokyo Autumn School --- Quantum Field Theory and Subfactors ---
    • Place of Presentation
      UC Berkeley
    • Year and Date
      2016-11-14
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] On quantum elliptic Algebras2016

    • Author(s)
      Yoshihisa Saito
    • Organizer
      Geometric representation theory
    • Place of Presentation
      京都大学理学部
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Remarks] Invariants of the Weyl group of type A_2l^(2)

    • URL

      https://arxiv.org/pdf/1903.00266.pdf

    • Related Report
      2018 Research-status Report

URL: 

Published: 2016-04-21   Modified: 2021-02-19  

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