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Quantized Schur algebras, Koszul duality and categorifications

Research Project

Project/Area Number 24740011
Research Category

Grant-in-Aid for Young Scientists (B)

Allocation TypeMulti-year Fund
Research Field Algebra
Research InstitutionOsaka City University

Principal Investigator

MIYACHI Hyohe  大阪市立大学, 大学院理学研究科, 准教授 (90362227)

Research Collaborator Chlouveraki Maria  
Chuang Joseph  
Fang Ming  
Leclerc Bernard  
Rouquier Raphael  
Tan Kai Meng  
KUWABARA Toshiro  
Project Period (FY) 2012-04-01 – 2017-03-31
Project Status Completed (Fiscal Year 2016)
Budget Amount *help
¥4,030,000 (Direct Cost: ¥3,100,000、Indirect Cost: ¥930,000)
Fiscal Year 2015: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2014: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2013: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2012: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Keywords表現論 / 量子群 / Hecke環 / 準遺伝的多元環 / 圏化 / 導来同値 / ヘッケ環 / 結晶基底 / 導来圏 / Hochschild コホモロジー / 国際研究者交流(仏,英,米) / 国際情報交換(仏,英,米) / Schur algebras / categorification / Representation Theory / Quantum Groups / Fock spaces / Hecke algebras / 国際研究者交流(アメリカ,イギリス,中国)
Outline of Final Research Achievements

About 100 years ago, Representation Theory launched. There are various mathematical research fields such as algebras, geometry and analysis, and Representation Theory interacts with them. One of the most important thing in High school chemistry is the notion of atom, which represents the smallest basic unit. In Representation Theory, the corresponding notion is the simple object. Moreover, molecules in High school chemistry are also important. In a sense, the corresponding notion of these in Representation Theory are projective indecomposable objects, which are extended as unbreakable units by simple objects as possible.
By standing on the shoulder of giants, we knew that those two kinds of objects are interpreted as two kinds of Lusztig's canonical bases also known as Kashiwara's global crystal bases in the research area on this report. Based on those interpretation we found in the category theory a natural interpretation on exchanging those two kinds of objects.

Report

(6 results)
  • 2016 Annual Research Report   Final Research Report ( PDF )
  • 2015 Research-status Report
  • 2014 Research-status Report
  • 2013 Research-status Report
  • 2012 Research-status Report
  • Research Products

    (12 results)

All 2016 2013 2012 Other

All Int'l Joint Research (6 results) Presentation (6 results) (of which Invited: 6 results)

  • [Int'l Joint Research] Department of Mathematics/City University of London(United Kingdom)

    • Related Report
      2016 Annual Research Report
  • [Int'l Joint Research] Department of Mathematics/National University of Singapore(シンガポールvvv)

    • Related Report
      2016 Annual Research Report
  • [Int'l Joint Research] Chinese Academy of Sciences/Beijing(中国)

    • Related Report
      2016 Annual Research Report
  • [Int'l Joint Research] Ming Fang/Chinese Academy of Science(中国)

    • Related Report
      2015 Research-status Report
  • [Int'l Joint Research] Joseph Chuang/City University, London(英国)

    • Related Report
      2015 Research-status Report
  • [Int'l Joint Research] Raphael Rouquier/UCLA(米国)

    • Related Report
      2015 Research-status Report
  • [Presentation] Level-Rank Dualiy2016

    • Author(s)
      宮地 兵衛
    • Organizer
      Summer School on Quasi-hereditary Algebras
    • Place of Presentation
      大阪府立大学 A5棟 124号室 大阪府堺市中区学園町
    • Related Report
      2016 Annual Research Report
    • Invited
  • [Presentation] Rational Cherednik algebras and finite Chevalley groups in characteristic l>02013

    • Author(s)
      宮地 兵衛
    • Organizer
      Modular Representation Theory of Finite and p-adic Groups
    • Place of Presentation
      シンガポール国立大学, シンガポール共和国
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] Cuspidal modules over rational Cherednik algebras and finite Chevalley groups2013

    • Author(s)
      宮地 兵衛
    • Organizer
      The Third International Symposium on Groups, Algebras and Related Topics
    • Place of Presentation
      北京大学, 中華人民共和国
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] Rational Cherednik algebras and finite Chevalley groups in characteristic l>02013

    • Author(s)
      宮地 兵衛
    • Organizer
      Modular Representation Theory of Finite and p-adic Groups
    • Place of Presentation
      シンガポール国立大学, シンガポール
    • Related Report
      2012 Research-status Report
    • Invited
  • [Presentation] TBA2013

    • Author(s)
      宮地 兵衛
    • Organizer
      The Third International Symposium on Groups, Algebras and Related Topics
    • Place of Presentation
      北京大学, 中華人民共和国
    • Related Report
      2012 Research-status Report
    • Invited
  • [Presentation] 一般線形群のモジュラー表現の圏の比較2012

    • Author(s)
      宮地 兵衛
    • Organizer
      日本数学会2012年秋季総合分科会 代数学分科会特別講演
    • Place of Presentation
      九州大学伊都キャンパス(福岡県)
    • Related Report
      2012 Research-status Report
    • Invited

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Published: 2013-05-31   Modified: 2022-02-16  

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