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Geometric study of quantum groups and associative algebras

Research Project

Project/Area Number 24540008
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

Saito Yoshihisa  東京大学, 数理(科)学研究科(研究院), 准教授 (20294522)

Co-Investigator(Renkei-kenkyūsha) NAITO Satoshi  東京工業大学, 理工学研究科, 教授 (60252160)
IYAMA Osamu  名古屋大学, 多元数理研究科, 教授 (70347532)
TANISAKI Toshiyuki  大阪市立大学, 理学研究科, 教授 (70142916)
Project Period (FY) 2012-04-01 – 2016-03-31
Project Status Completed (Fiscal Year 2015)
Budget Amount *help
¥5,200,000 (Direct Cost: ¥4,000,000、Indirect Cost: ¥1,200,000)
Fiscal Year 2015: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2014: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2013: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2012: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Keywords結晶基底 / 量子群
Outline of Final Research Achievements

The main results of this research are the followings.
(1) Recently, Kamnitzer gave a new realization of crystal bases of the quantized enveloping algebras of finite types, by using the theory of Mirkovic-Vilinen polytopes which has originated in geometry of affine Grassmannian varieties. In this research, we constructed an analogue of these theories for affine quantized enveloping algebras.
(2) We studied characteristic varieties of intersection cohomology complexes of Schubert varieties in type A. More explicitly, by using the method of geometric realization of crystal cases, we determined the explicit forms of these varieties in low rank cases.

Report

(5 results)
  • 2015 Annual Research Report   Final Research Report ( PDF )
  • 2014 Research-status Report
  • 2013 Research-status Report
  • 2012 Research-status Report
  • Research Products

    (16 results)

All 2016 2015 2014 2013 2012 Other

All Journal Article (4 results) (of which Int'l Joint Research: 1 results,  Peer Reviewed: 4 results,  Open Access: 1 results,  Acknowledgement Compliant: 1 results) Presentation (12 results) (of which Int'l Joint Research: 2 results,  Invited: 7 results)

  • [Journal Article] Quantized coordinate rings, PBW-type bases and q-boson algebras2016

    • Author(s)
      Yoshihisa Saito
    • Journal Title

      Journal od Algebra

      Volume: 453 Pages: 456-491

    • Related Report
      2015 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] Toward Berenstein-Zelevinsky data in affine type A, part III2013

    • Author(s)
      S. Naito, D. Sagaki, and Y. Saito
    • Journal Title

      Proof of the connectedness, Springer Proceedings in Mathematics and Statistics

      Volume: Vol.40 Pages: 361-402

    • DOI

      10.1007/978-1-4471-4863-0_15

    • ISBN
      9781447148623, 9781447148630
    • Related Report
      2013 Research-status Report 2012 Research-status Report
    • Peer Reviewed
  • [Journal Article] Toward Berenstein-Zelevinsky data in affine type A, Part I : Construction of the affine analogs2012

    • Author(s)
      S.Naito, D.Sagaki, Y. Saito
    • Journal Title

      Contemporary Mathematics

      Volume: 565 Pages: 143-184

    • DOI

      10.1090/conm/565/11180

    • Related Report
      2012 Research-status Report
    • Peer Reviewed
  • [Journal Article] Toward Berenstein-Zelevinsky data in affine type $A$, I: Construction of affine analogs2012

    • Author(s)
      Daisuke Sagaki, Satoshi Naito, and Yoshihisa Saito
    • Journal Title

      http://dx.doi.org/10.1090/conm/565/11187

      Volume: 565 Pages: 143-184

    • Related Report
      2012 Research-status Report
    • Peer Reviewed
  • [Presentation] 表現論 vs. 表現論2016

    • Author(s)
      Yoshihisa Saito
    • Organizer
      代数学若手会
    • Place of Presentation
      奈良女子大学
    • Year and Date
      2016-03-29
    • Related Report
      2015 Annual Research Report
    • Invited
  • [Presentation] 楕円ルート系について2016

    • Author(s)
      Yoshihisa Saito
    • Organizer
      信州代数セミナー
    • Place of Presentation
      信州大学
    • Year and Date
      2016-02-18
    • Related Report
      2015 Annual Research Report
    • Invited
  • [Presentation] On Elliptic Artin Groups2015

    • Author(s)
      Yoshihisa Saito
    • Organizer
      Shanghai Conference on Representation Theory
    • Place of Presentation
      Tian He Hotel (Shanghai,China)
    • Year and Date
      2015-12-11
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Geometric construction of crystal bases and its applications2015

    • Author(s)
      Yoshihisa Saito
    • Organizer
      Series of Lectures on representation theory
    • Place of Presentation
      Tongji University (China)
    • Year and Date
      2015-10-25
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Quantized coordinate rings, PBW bases and $q$-boson algebras2014

    • Author(s)
      Yoshihisa Saito
    • Organizer
      Shanghai Workshop on Representation Theory
    • Place of Presentation
      Tongji University, Shanghai,China
    • Year and Date
      2014-12-08
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] 量子座標環とその表現2014

    • Author(s)
      斉藤義久
    • Organizer
      第17回代数群と量子群の表現論
    • Place of Presentation
      呉羽ハイツ, 富山県富山市
    • Year and Date
      2014-08-08
    • Related Report
      2014 Research-status Report
  • [Presentation] Quantized coordinate rings, PBW bases and $q$-boson algebras2014

    • Author(s)
      Yoshihisa Saito
    • Organizer
      Representation Theory and Related Topics (ICM2014 Satellite Conference)
    • Place of Presentation
      EXCO, Daegu, Korea
    • Year and Date
      2014-08-08
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] 量子座標環の表現論

    • Author(s)
      斉藤義久
    • Organizer
      大阪市立大学談話会
    • Place of Presentation
      大阪市立大学
    • Related Report
      2013 Research-status Report
  • [Presentation] Schubert多様体のIC complexの特性多様体に関する一観察

    • Author(s)
      斉藤義久
    • Organizer
      大阪表現論セミナー
    • Place of Presentation
      大阪市立大学文化交流センター
    • Related Report
      2013 Research-status Report
  • [Presentation] Realization of crystal bases

    • Author(s)
      Yoshihisa Saito
    • Organizer
      The 2nd mini-symposium on Representation theory
    • Place of Presentation
      Jeju (Korea)
    • Related Report
      2012 Research-status Report
    • Invited
  • [Presentation] Toward Berenstein-Zelevinsky data in affine type $A$

    • Author(s)
      Yoshihisa Saito
    • Organizer
      代数解析学とその周辺
    • Place of Presentation
      早稲田大学理工学部
    • Related Report
      2012 Research-status Report
  • [Presentation] Toward Berenstein-Zelevinsky data in affine type $A$

    • Author(s)
      Yoshihisa Saito
    • Organizer
      第15回代数群と量子群の表現論
    • Place of Presentation
      いこいの村アゼィリア飯綱
    • Related Report
      2012 Research-status Report

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Published: 2013-05-31   Modified: 2023-03-16  

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