• Search Research Projects
  • Search Researchers
  • How to Use
  1. Back to previous page

Geometric study of quantum groups and its application to representation theory of algebras

Research Project

Project/Area Number 20540009
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionThe University of Tokyo

Principal Investigator

SAITO Yoshihisa  東京大学, 大学院・数理科学研究科, 准教授 (20294522)

Co-Investigator(Renkei-kenkyūsha) IYAMA Osamu  名古屋大学, 大学院・多元数理研究科, 教授 (70347532)
SAITO Kyoji  東京大学, 数物連携宇宙機構, 主任研究員 (20012445)
TANISAKI Toshiyuki  大阪市立大学, 理学研究科, 教授 (70142916)
NAITO Satoshi  東京工業大学, 大学院・理工学研究科, 教授 (60252160)
Project Period (FY) 2008 – 2011
Project Status Completed (Fiscal Year 2011)
Budget Amount *help
¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000)
Fiscal Year 2011: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2010: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2009: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2008: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Keywords量子群 / 結晶基底
Research Abstract

Around 2000, Mirkovic and Vilonen defined a new class of algebraic cycles in affine Grassmanians, called Mirkovic-Vilonen(MV for short) cycles. By the definition, these cycles has an action of a real maximal Torus, and their moment map image are plopytoles in a real Cartan subalgebra, called MV polytopes. After their work, Kamnitzer defined a crstal structure on the set of all MV polytopes, and proved that it is isomorphic to the crystal basis of the negative half of quantum universal enveloping algebras of finite type. Moreover, it is known that the followings are eqivalent : to give a MV polytope, and to give a collection of nonnegative integer, called Berenstein-Zelevinsky(BZ for short) data. In other words, he introduced a new realization of the crystal basis of the negative half of quantum universal enveloping algebras of finite type, by using BZ data.
In this study, we generalize his result to the case of affine type A. More precisely, we define a notion of affine BZ data, and prove that the set of all affine BZ data has a crystal structure which are isomorphic to the negative half of quantum universal enveloping algebras of affine type A. Namely, we get a new realization of the crystal basis of the negative half of quantum universal enveloping algebras of affine type A, by using affine BZ data. Our main results are stated by combinatorial language, but in our proof, a geometric construction of crystal basis due to Kashiwara and the author plays a crucial role.

Report

(6 results)
  • 2011 Annual Research Report   Final Research Report ( PDF )
  • 2010 Annual Research Report   Self-evaluation Report ( PDF )
  • 2009 Annual Research Report
  • 2008 Annual Research Report
  • Research Products

    (49 results)

All 2012 2011 2010 2009 2008 Other

All Journal Article (22 results) (of which Peer Reviewed: 20 results) Presentation (24 results) Book (3 results)

  • [Journal Article] Towoard Berenstein Zalevinsky data in affine type A, II : Expricit description2012

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

      Contemp. Math

      Volume: 565 Pages: 185-216

    • Related Report
      2011 Final Research Report
    • Peer Reviewed
  • [Journal Article] Towoard Berenstein Zalevinsky data in affine type A, I : Constriction of affine analogs2012

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

      Contemp. Math

      Volume: 565 Pages: 143-184

    • Related Report
      2011 Final Research Report
    • Peer Reviewed
  • [Journal Article] Mirkovic Vilonen polytopes and a quiver construction of crystal basis in type A2012

    • Author(s)
      Yoshihisa Saito
    • Journal Title

      Int. Math. Res. Notices

    • Related Report
      2011 Final Research Report
    • Peer Reviewed
  • [Journal Article] Toward Berenstein-Zelevinsky data in affine type A, I : Construction ogaffne analogs2012

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

      Contemp.Math.

      Volume: 565 Pages: 143-184

    • Related Report
      2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Toward Berenstein-Zelevinsky data in affine type A, II : Explicit description2012

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

      Contemp.Math.

      Volume: 565 Pages: 185-216

    • Related Report
      2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Indecompozable decomposition of tensor products of modules over the restricted quantum universal enveloping algebras associated to sl_22011

    • Author(s)
      Hiroki Kondo, Yoshihisa Saito
    • Journal Title

      J. Alg.

      Volume: 330 Pages: 103-129

    • Related Report
      2011 Final Research Report
    • Peer Reviewed
  • [Journal Article] Mirkovic-Vilonen polytoples and a quiver construction of crystal basis in type A2011

    • Author(s)
      Yoshihisa Saito
    • Journal Title

      Int.Math.Res.Notices

      Volume: 2012 Issue: 17 Pages: 3877-3928

    • DOI

      10.1093/imrn/rnr173

    • Related Report
      2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Indecomposable decomposition of tensor products of modules over the restricted quantum enveloping algebra associated to $sl_2$2011

    • Author(s)
      Yoshihisa SAITO, Hiroki KONDO
    • Journal Title

      Journal of Algbera 330

      Pages: 103-129

    • Related Report
      2010 Self-evaluation Report
    • Peer Reviewed
  • [Journal Article] Indecomposable decomposition of tensor products of modules over the restricted quantum universal enveloping algebra associated to $s1_2$2011

    • Author(s)
      Hiroki KONDO, Yoshihisa SAITO
    • Journal Title

      Journal of Algebra

      Volume: 330 Pages: 103-129

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Double affine Hecke algebras and elliptic Hecke algebras2010

    • Author(s)
      Yoshihisa Saito, Midori Shiota
    • Journal Title

      Progress in Math.

      Volume: 284 Pages: 297-312

    • Related Report
      2011 Final Research Report
    • Peer Reviewed
  • [Journal Article] On Hecke algebras associated with elliptic root systems2010

    • Author(s)
      Yoshihisa SAITO, Midori SHIOTA
    • Journal Title

      Representation Theory of Algebraic Groups and Quantum Groups (Progress in Math.)

      Volume: 284 Pages: 297-312

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Yoshihiro Takeyama, The rational qKZ equation and shifted non-symmetric Jack polynomials2009

    • Author(s)
      Saburo Kakei, Michitomo Nishizawa, Yoshihisa Saito
    • Journal Title

      SIGMA

      Volume: 5 Pages: 12-12

    • Related Report
      2011 Final Research Report
    • Peer Reviewed
  • [Journal Article] On Hecke algebras associated with elliptic root systems and the double affine Hecke algebras, Publ2009

    • Author(s)
      Yoshihisa Saito, Midori Shiota
    • Journal Title

      Res. Inst. Math. Sci.

      Volume: 45 Pages: 845-905

    • Related Report
      2011 Final Research Report
    • Peer Reviewed
  • [Journal Article] The rational qKZ equation and shifted non-symmetric Jack polynomials2009

    • Author(s)
      Saburo KAKEI, Michitomo NISHIZAWA, Yoshihisa SAITO, Yoshihiro TAKEYAMA
    • Journal Title

      SGMA 5 10

      Pages: 12-12

    • Related Report
      2010 Self-evaluation Report
    • Peer Reviewed
  • [Journal Article] On Hecke algebras associated with elliptic root systems and the double affine Hecke algebras2009

    • Author(s)
      Yoshihisa SAITO, Midori SHIOTA
    • Journal Title

      Publ.RIMS. 45

      Pages: 845-905

    • Related Report
      2010 Self-evaluation Report
    • Peer Reviewed
  • [Journal Article] On Hecke algebras associated with elliptic root systems2009

    • Author(s)
      Yoshihisa SAITO, Midori SHIOTA
    • Journal Title

      Progress in Math. (to appear, 掲載確定)

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On Hecke algebras associated with elliptic root systems and the double affine Hecke algebras2009

    • Author(s)
      Yoshihisa SAITO, Midori SHIOTA
    • Journal Title

      Publications RIMS. 45

      Pages: 845-905

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On tensor products of Mirkovic-Vilonen polytopes in type A2009

    • Author(s)
      Satoshi NAITO, Daisuke SAGAKI, Yoshihisa SAITO
    • Journal Title

      研究集会「表現論と組合せ論」報告集 (to appear, 掲載確定)

    • Related Report
      2009 Annual Research Report
  • [Journal Article] On Hecke algebras associated with elliptic root systems2009

    • Author(s)
      Yoshihisa SAITO and Midori SHIOTA
    • Journal Title

      Progress in Math.(to appear)

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On Hecke algebras associated with elliptic root systems and the double affine Hecke algebras2009

    • Author(s)
      Yoshihisa SAITO and Midori SHIOTA
    • Journal Title

      Publ. RIMS (to appear)

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] The rational qKZ equation and shifted non-symmetric Jackpolynomials2008

    • Author(s)
      S. KAKEI, M. NISHIZAWA, Y. SAITO and Y.TAKEYAMA
    • Journal Title

      SIGMA 5

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Berenstein-Zelevinsky data and the crystal basis of $U_q^-$ in type $A_n^{(1)}$

    • Author(s)
      Yoshihisa SAITO
    • Journal Title

      数理解析研究所講究録「組合せ論的表現論とその応用」(to appear)

      Volume: (掲載確定)

    • Related Report
      2010 Annual Research Report
  • [Presentation]2011

    • Author(s)
      Yoshihisa Saito
    • Organizer
      Symmetries, Integrable systems and Representations
    • Place of Presentation
      Universite 1(Lyon,フランス)(招待講演)
    • Year and Date
      2011-12-16
    • Related Report
      2011 Annual Research Report
  • [Presentation] On Berenstein-Zelevinsky data in affine type A2011

    • Author(s)
      Yoshihisa Saito
    • Organizer
      Conformed Field Theories and Tensor Categories
    • Place of Presentation
      Peking Unoversity(北京,中国)(招待講演)
    • Year and Date
      2011-05-13
    • Related Report
      2011 Annual Research Report
  • [Presentation] On Berenstein Zelevinsky data in affine type A, Symmetries2011

    • Author(s)
      Yoshihisa Saito
    • Organizer
      Integrable systems and Representations
    • Place of Presentation
      Universite 1, Lyon, France
    • Related Report
      2011 Final Research Report
  • [Presentation] 前射影多元環と量子群の結晶基底2011

    • Author(s)
      斉藤義久
    • Organizer
      第44回環論および表現論シンポジウム
    • Place of Presentation
      岡山大学理学部
    • Related Report
      2011 Final Research Report
  • [Presentation] On Berenstein Zelevinsky data in affine type A, Conformal field theories and tensor categories2011

    • Author(s)
      Yoshihisa Saito
    • Organizer
      Beiging International center for Mathematical Research
    • Place of Presentation
      Beiging, China
    • Related Report
      2011 Final Research Report
  • [Presentation] 前射影多元環と量子群の結晶基底2011

    • Author(s)
      斉藤義久
    • Organizer
      第44回環論および表現論シンポジウム
    • Place of Presentation
      岡山大学理学部(招待講演)(連続講演)
    • Related Report
      2011 Annual Research Report
  • [Presentation] Bernstein-Zelevinsky data and the crystal basis of $U_q^-$ in type $A_{n-1}^{(1)}$2010

    • Author(s)
      Yoshihisa SAITO
    • Organizer
      組合せ論的表現論とその応用
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2010-10-22
    • Related Report
      2010 Annual Research Report
  • [Presentation] Mirkovic-Vilonen polytopes and quiver construction of crystal basis in type $A_n$2010

    • Author(s)
      Yoshihisa SAITO
    • Organizer
      Representation Theory of Algebraic Groups and Quantum Groups 10
    • Place of Presentation
      Nagoya University (Nagoya, Japan)
    • Year and Date
      2010-08-03
    • Related Report
      2010 Annual Research Report
  • [Presentation] On tensor category arising from representation theory of the restricted quantum universal enveloping algebra associated to $s1_2$2010

    • Author(s)
      Yoshihisa SAITO
    • Organizer
      Interplay between representation theory and Geometry
    • Place of Presentation
      Tsinghua University (Beijing, China)
    • Year and Date
      2010-05-04
    • Related Report
      2010 Annual Research Report
  • [Presentation] Crystals, MV polytopes and tropical flag varieties2010

    • Author(s)
      Yoshihisa SAITO
    • Organizer
      トロピカル幾何と超離散系の新展開
    • Place of Presentation
      京都大学大学院工学研究科
    • Year and Date
      2010-03-11
    • Related Report
      2009 Annual Research Report
  • [Presentation] Berenstein-Zelevonsky data and the crystal basis of U_q^-in tyle A_{ n-1}^{(1)}2010

    • Author(s)
      斉藤義久
    • Organizer
      組合せ論的表現論とその応用
    • Place of Presentation
      京都大学数理解析研究所
    • Related Report
      2011 Final Research Report
  • [Presentation] Mirkovic-Vilonen polytopes and a quiver construction of crystal basis in type A2010

    • Author(s)
      Yoshihisa Saito
    • Organizer
      Representation Theory og Algebraic Groups and Quantum Groups' 10
    • Place of Presentation
      Nagoya University
    • Related Report
      2011 Final Research Report
  • [Presentation] On tensor category arising from representation theory of the restricted quantum universal enveloping algebra associated to sl_22010

    • Author(s)
      Yoshihisa Saito
    • Organizer
      Interplay between representation theory and geometry
    • Place of Presentation
      Tsinghua University Beijing, China
    • Related Report
      2011 Final Research Report
  • [Presentation] Crystals, MV polytopes and' toropical flag varieties'2010

    • Author(s)
      斉藤義久
    • Organizer
      トロピカル幾何と超離散系の新展開
    • Place of Presentation
      京都大学大学院工学研究科
    • Related Report
      2011 Final Research Report
  • [Presentation] On tensor category arising form representation theory of the restricted quantum universal enveloping algebra associated to $sl_2$2010

    • Author(s)
      Yoshisa SAITO
    • Organizer
      Invited talk, Interplay between representation theory and geomerry
    • Place of Presentation
      Tsinghua University (Beijing, China)
    • Related Report
      2010 Self-evaluation Report
  • [Presentation] Mirkovic-Vilonen polytopes and quiver construction of crystal basis in type $A$2010

    • Author(s)
      Yoshihisa SAITO
    • Organizer
      Invited talk, Representation theory og Algberaic Groups and Quantum Groups '10
    • Place of Presentation
      Nagoya University (Nagoya, Japan)
    • Related Report
      2010 Self-evaluation Report
  • [Presentation] On tensor category arising from representation theory of the restricted quantum universal enveloping algebra associated to s1\22009

    • Author(s)
      Yoshihisa SAITO
    • Organizer
      International workshop on combinatorial and geometric approach to representation theory
    • Place of Presentation
      Seoul National University(Seoul, Korea)
    • Year and Date
      2009-09-23
    • Related Report
      2009 Annual Research Report
  • [Presentation] On tensor products of Mirkovic-Vilonen polytopes in type A2009

    • Author(s)
      Yoshihisa SAITO
    • Organizer
      表現論と組合せ論
    • Place of Presentation
      北海道大学大学院理学研究科
    • Year and Date
      2009-08-26
    • Related Report
      2009 Annual Research Report
  • [Presentation] On tensor category arising from representation theory of the restricted quantum universal enveloping algebra associated to sl_22009

    • Author(s)
      Yoshihisa Saito
    • Organizer
      International workshop on combinatorial and geometric approach to representation theory
    • Place of Presentation
      Seoul National University Seoul, Korea
    • Related Report
      2011 Final Research Report
  • [Presentation] Small quantum group U_q(sl_2)の表現のテンソル積の直既約分解について2009

    • Author(s)
      斉藤義久
    • Organizer
      日本数学会2009年度年会
    • Place of Presentation
      東京大学
    • Related Report
      2011 Final Research Report
  • [Presentation] On tensor category arising form representation theory of the restricted quantum universal enveloping algebra associated to $sl_2$2009

    • Author(s)
      Yoshihisa SAITO
    • Organizer
      Invited talk, International workshop on combinatorial and geometric representation theory
    • Place of Presentation
      Seoul National University (Seoul, Korea)
    • Related Report
      2010 Self-evaluation Report
  • [Presentation] Restricted quantum group U_q(sl_2)の表現のテンソル積の直既約分解について2008

    • Author(s)
      斉藤義久
    • Organizer
      日本数学会年会
    • Place of Presentation
      東京大学
    • Year and Date
      2008-03-28
    • Related Report
      2008 Annual Research Report
  • [Presentation] Hecke代数の多項式表現について2008

    • Author(s)
      斉藤義久
    • Organizer
      第53回代数学シンポジウム
    • Place of Presentation
      盛岡駅前アイーナ
    • Related Report
      2011 Final Research Report
  • [Presentation] Hecke代数の多項式表現について2008

    • Author(s)
      Yoshihisa SAITO
    • Organizer
      第53回代数学シンポジウム盛岡駅前アリーナ
    • Place of Presentation
      招待講演
    • Related Report
      2010 Self-evaluation Report
  • [Book] Algebraic Analysis and Around2009

    • Publisher
      Math. Soc
    • Related Report
      2011 Final Research Report
  • [Book] Advanced studies in Pure Mathematics(Algebraic analysis and around (編集)2009

    • Author(s)
      Atsuhi MATSUO, Tetsuji MIWA, Toshiki NAKASHIMA, Yoshihisa SAITO
    • Total Pages
      54
    • Publisher
      Math.Soc.Japan
    • Related Report
      2010 Self-evaluation Report
  • [Book] Algebraic Analysis and Around2009

    • Author(s)
      Atsushi MATSUO, Tetsuji MIWA, Toshiki NAKASHIMA, Yoshihisa SAITO(編集)
    • Total Pages
      379
    • Publisher
      日本数学会
    • Related Report
      2009 Annual Research Report

URL: 

Published: 2008-04-01   Modified: 2016-04-21  

Information User Guide FAQ News Terms of Use Attribution of KAKENHI

Powered by NII kakenhi