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Geometric study of algebras attached to root systems

Research Project

Project/Area Number 26287004
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypePartial Multi-year Fund
Section一般
Research Field Algebra
Research InstitutionKyoto University

Principal Investigator

Kato Syu  京都大学, 理学研究科, 准教授 (40456760)

Project Period (FY) 2014-04-01 – 2019-03-31
Project Status Completed (Fiscal Year 2018)
Budget Amount *help
¥16,250,000 (Direct Cost: ¥12,500,000、Indirect Cost: ¥3,750,000)
Fiscal Year 2018: ¥3,120,000 (Direct Cost: ¥2,400,000、Indirect Cost: ¥720,000)
Fiscal Year 2017: ¥3,120,000 (Direct Cost: ¥2,400,000、Indirect Cost: ¥720,000)
Fiscal Year 2016: ¥3,120,000 (Direct Cost: ¥2,400,000、Indirect Cost: ¥720,000)
Fiscal Year 2015: ¥3,120,000 (Direct Cost: ¥2,400,000、Indirect Cost: ¥720,000)
Fiscal Year 2014: ¥3,770,000 (Direct Cost: ¥2,900,000、Indirect Cost: ¥870,000)
Keywords幾何学的拡大代数 / 半無限旗多様体 / affine Hecke代数 / 箙Hecke代数 / 一般Springer対応 / アフィン・グラスマン多様体 / コストカ多項式 / 非対称Macdonald多項式 / 非対称マクドナルド多項式 / レベル制限コストカ多項式 / アフィン旗多様体 / 共型場理論 / アフィングラスマン多様体 / カレント代数 / 共形場理論 / 柏原旗多様体 / Demazure加群 / KLR代数 / 大域ワイル加群 / ピエリ公式 / 代数群 / 表現論 / 旗多様体 / スプリンガー対応
Outline of Final Research Achievements

In the first half of the research period, we have developed and polished the theory of geometric extension algebras from the both of the general theory and examples like quiver Hecke algebras and generalized Springer correspondence. In particulcar, we have revealed that quiver Hecke algebras possesses a structure similar to the classical theory of highest weight categories, and find that the orthogonality relation of Green functions arising from representation theory of Chevalley groups are direct corollaries of some orthogonality in the sense of homological algebras. This resolves several conjectures in this area.

In the latter half, we have studied the representation theory of current algebras and geometry of semi-infinite flag manifolds and affine Grassmanians. Although the setting is different, the pattern is similar here. Consequently we have proved several conjectures also in this area.

Academic Significance and Societal Importance of the Research Achievements

表現論とは(群などの)対称性を固定してその実現がどの程度あるか分類し、それらの間の関係を研究する数学分野である。古典的には表現論が半単純、つまり任意の実現が原始的なもののの集まりとしてかける状況が大切であった(例えば、素粒子の分類などは実際にそのような現象と結びついている)。しかし、現実が素粒子や原子の単純な集まりとは異なり互いに干渉しあうように対称性も単純な集まりの間に相互関係がある場合が大切であることが分かってきた。本研究の成果はそのような相互関係がある対称性の理論を今までより一歩推し進め、異なる場所の間の相互関係も許すようなものを許容すると古典的な対称性もよりよくわかるというものである。

Report

(6 results)
  • 2018 Annual Research Report   Final Research Report ( PDF )
  • 2017 Annual Research Report
  • 2016 Annual Research Report
  • 2015 Annual Research Report
  • 2014 Annual Research Report
  • Research Products

    (26 results)

All 2019 2018 2017 2016 2015 Other

All Int'l Joint Research (5 results) Journal Article (8 results) (of which Int'l Joint Research: 2 results,  Peer Reviewed: 8 results,  Acknowledgement Compliant: 3 results) Presentation (8 results) (of which Int'l Joint Research: 4 results,  Invited: 7 results) Remarks (3 results) Funded Workshop (2 results)

  • [Int'l Joint Research] Higher School of Economics(ロシア連邦)

    • Related Report
      2018 Annual Research Report
  • [Int'l Joint Research] Higher School of Economics(ロシア連邦)

    • Related Report
      2017 Annual Research Report
  • [Int'l Joint Research] Virginia Tech.(米国)

    • Related Report
      2017 Annual Research Report
  • [Int'l Joint Research] Higher School of Mathematics(ロシア連邦)

    • Related Report
      2016 Annual Research Report
  • [Int'l Joint Research] Max-Planck Institute fur Mathematik(Germany)

    • Related Report
      2016 Annual Research Report
  • [Journal Article] Representation theoretic realizations of non-symmetric Macdonald polynomials at infinity2019

    • Author(s)
      Evgeny Feigin, 加藤周, Ievgen Makedonskyi
    • Journal Title

      Journal fuer die reine und angewandte Mathematik

      Volume: 受理済み

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Frobenius splitting of thick flag manifolds of Kat-Moody algebras2018

    • Author(s)
      加藤周
    • Journal Title

      International Mathematics Research Notices

      Volume: 印刷中 Issue: 17 Pages: 5401-5427

    • DOI

      10.1093/imrn/rny174

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On the monoidality of Saito reflection functors2018

    • Author(s)
      加藤周
    • Journal Title

      International Mathematics Research Notices

      Volume: 印刷中 Issue: 22 Pages: 8600-8623

    • DOI

      10.1093/imrn/rny233

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A Weyl module filtration of an integrable representations2018

    • Author(s)
      加藤周, Sergey Loktev
    • Journal Title

      Communications in Mathematical Physics

      Volume: 印刷中

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Demazure character formula for semi-infinite flag varieties2018

    • Author(s)
      加藤周
    • Journal Title

      Mathematische Annalen

      Volume: 印刷中 Issue: 3-4 Pages: 1769-1801

    • DOI

      10.1007/s00208-018-1652-5

    • NAID

      120006491829

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed
  • [Journal Article] An algebraic study of extension algebras2017

    • Author(s)
      Syu Kato
    • Journal Title

      American Journal of Mathematics

      Volume: 139 no.3

    • NAID

      120006410740

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] A homological study of Green polynomials2015

    • Author(s)
      Syu Kato
    • Journal Title

      Annales scientifiques de l'ENS

      Volume: 48 Pages: 1035-1074

    • Related Report
      2015 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] A homological study of Green polynomials2015

    • Author(s)
      Syu Kato
    • Journal Title

      Annals Scientifiques de l'Ecole Normale Supreme

      Volume: 48

    • Related Report
      2014 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Presentation] Frobenius splitting of semi-infinite flag manifolds2019

    • Author(s)
      加藤周
    • Organizer
      Taipei conference in representation theory VI
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Frobenius splitting of semi-infinite flag manifolds2019

    • Author(s)
      加藤周
    • Organizer
      Representation theory, gauge theory, and integrable systems
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Loop structure on equivariant $K$-theory of semi-infinite flag manifolds2018

    • Author(s)
      加藤周
    • Organizer
      GEOMETRY AND REPRESENTATION THEORY AT THE INTERFACE OF LIE ALGEBRAS AND QUIVERS
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Equivariant K-theory of semi-infinite flag manifolds and quantum K-theory of flag manifolds2018

    • Author(s)
      加藤周
    • Organizer
      Quantum K-theory and related topics
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] An algebraic study of extension algebras2017

    • Author(s)
      Syu Kato
    • Organizer
      Representation Theory Seminar
    • Place of Presentation
      UC Chapell Hill
    • Year and Date
      2017-04-15
    • Related Report
      2016 Annual Research Report
  • [Presentation] 半無限旗多様体の幾何学と表現論2017

    • Author(s)
      加藤周
    • Organizer
      Algebraic Lie Theory and Representation Theory 2017
    • Related Report
      2017 Annual Research Report
    • Invited
  • [Presentation] An algebraic study of extension algebras2015

    • Author(s)
      Syu Kato
    • Organizer
      Symplectic representation theory
    • Place of Presentation
      University of Glasgow
    • Year and Date
      2015-06-12
    • Related Report
      2015 Annual Research Report
    • Invited
  • [Presentation] 量子群やヘッケ環から生じるアフィン最高ウェイト圏とその応用について2015

    • Author(s)
      加藤周
    • Organizer
      日本数学会年会
    • Place of Presentation
      明治大学リバティホール
    • Year and Date
      2015-03-23
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Remarks] http://www.math.kyoto-u.ac.jp/~syuchan

    • Related Report
      2018 Annual Research Report
  • [Remarks] Home page of KATO, Syu

    • URL

      http://www.math.kyoto-u.ac.jp/~syuchan

    • Related Report
      2016 Annual Research Report
  • [Remarks]

    • URL

      http://www.math.kyoto-u.ac.jp/~syuchan

    • Related Report
      2014 Annual Research Report
  • [Funded Workshop] Representation theory of reductive Lie groups and algebras2019

    • Related Report
      2018 Annual Research Report
  • [Funded Workshop] Winter School on Representation Theory 20162016

    • Place of Presentation
      東京大学数理科学研究科
    • Year and Date
      2016-01-08
    • Related Report
      2015 Annual Research Report

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Published: 2014-04-04   Modified: 2022-02-22  

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