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作用素環と量子群

Research Project

Project/Area Number 14J02598
Research Category

Grant-in-Aid for JSPS Fellows

Allocation TypeSingle-year Grants
Section国内
Research Field Basic analysis
Research InstitutionThe University of Tokyo

Principal Investigator

荒野 悠輝  東京大学, 数理科学研究科, 特別研究員(DC1)

Project Period (FY) 2014-04-25 – 2017-03-31
Project Status Completed (Fiscal Year 2016)
Budget Amount *help
¥2,800,000 (Direct Cost: ¥2,800,000)
Fiscal Year 2016: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2015: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2014: ¥1,000,000 (Direct Cost: ¥1,000,000)
Keywords量子群 / テンソル圏 / 性質(T) / Howe-Moore性 / 作用素環 / q変形 / KK理論 / Drinfeld double
Outline of Annual Research Achievements

離散群上の正定値関数の研究は,幾何群論をはじめとして、作用素環,表現論,エルゴード理論など様々な研究において非常に重要な役割を果たしてきた.その量子群の類似として,De Commer-Freslon-Yamashitaによる離散量子群の中心的関数による近似性質/剛性,および等価な概念であるPopa-Vaes,Neshveyev-Yamashita,Ghosh-Jonesらによるテンソル圏の関数による近似性質/剛性は,部分因子環論との関連も含めて,重要な分野となってきた.
私は,昨年度に引き続き,このような中心的近似性/剛性について調べた.特に,q-変形のDrinfeld doubleの表現論を推し進めることによって,私の以前の結果であるSUq(2n+1)の表現圏の性質(T)を拡張し,一般の高階の単純コンパクトLie群のq-変形の表現圏が性質(T)を持つことを示した.また,SUq(n)の場合には,より強く,中心的正定値関数(あるいは同じことであるが,その表現圏上の正定値関数)をSL(n,C)上の正定値関数を用いて表示することができた.
また,正定値関数を拡張し,離散量子群上の中心的完全有界乗関数や,テンソル圏上の完全有界乗関数を調べた.このような中心的完全有界乗関数に関しては,De Commer-Freslon-YamashitaによるSUq(2)の中心的CMAPしか知られていなかったが,量子群の中心的Howe-Moore性および,テンソル圏のHowe-Moore性を導入し,q-変形についてこの性質を示した.これは中心的性質(T*)などの完全有界乗関数の剛性につながることが期待される.

Research Progress Status

28年度が最終年度であるため、記入しない。

Strategy for Future Research Activity

28年度が最終年度であるため、記入しない。

Report

(3 results)
  • 2016 Annual Research Report
  • 2015 Annual Research Report
  • 2014 Annual Research Report
  • Research Products

    (13 results)

All 2017 2016 2015 2014

All Journal Article (3 results) (of which Peer Reviewed: 3 results,  Acknowledgement Compliant: 3 results,  Open Access: 2 results) Presentation (10 results) (of which Int'l Joint Research: 4 results,  Invited: 6 results)

  • [Journal Article] Comparison of unitary duals of Drinfeld doubles and complex semisimple Lie groups2017

    • Author(s)
      Y. Arano
    • Journal Title

      Communications in Mathematical Physics

      Volume: 351 Issue: 3 Pages: 1137-1147

    • DOI

      10.1007/s00220-016-2704-x

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Unitary spherical representations of Drinfeld doubles2016

    • Author(s)
      Yuki Arano
    • Journal Title

      Journal fuer die reine und angewandte Mathematik

      Volume: 印刷中 Issue: 742 Pages: 157-186

    • DOI

      10.1515/crelle-2015-0079

    • Related Report
      2015 Annual Research Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Journal Article] C*-tensor categories and subfactors for totally disconnected groups2016

    • Author(s)
      Yuki Arano
    • Journal Title

      Proceedings of the 2015 Abel Symposium

      Volume: 印刷中

    • Related Report
      2015 Annual Research Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Presentation] Unitary representations of Drinfeld doubles2016

    • Author(s)
      Y. Arano
    • Organizer
      Workshop "Von Neumann Algebras"
    • Place of Presentation
      The Hausdorff Research Institute for Mathematics(ドイツ・ボン)
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Torsion-freeness for fusion rings and tensor C*-categories2016

    • Author(s)
      Y. Arano
    • Organizer
      第51回関数解析研究会
    • Place of Presentation
      草津セミナーハウス(群馬県・吾妻郡草津町)
    • Related Report
      2016 Annual Research Report
  • [Presentation] Representation theory of Drinfeld doubles2016

    • Author(s)
      Y. Arano
    • Organizer
      作用素環論の最近の進展
    • Place of Presentation
      京都大学RIMS(京都府・京都市)
    • Related Report
      2016 Annual Research Report
    • Invited
  • [Presentation] Representation theory of Drinfeld doubles2016

    • Author(s)
      Y. Arano
    • Organizer
      Topological quantum groups and Hops algebras
    • Place of Presentation
      IMPAN(ポーランド・ワルシャワ)
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Unitary representations of Drinfeld doubles2015

    • Author(s)
      Yuki Arano
    • Organizer
      Facets of quantum homogeneous spaces
    • Place of Presentation
      Vrije Universiteit Brussel・ブリュッセル(ベルギー)
    • Year and Date
      2015-12-03
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Diagrammatic method for representation theory of complex semisimple (quantum) groups2015

    • Author(s)
      Yuki Arano
    • Organizer
      第50回関数解析研究会
    • Place of Presentation
      軽井沢町中央公民館(長野県軽井沢市)
    • Year and Date
      2015-09-03
    • Related Report
      2015 Annual Research Report
  • [Presentation] Unitary spherical representations of Drinfeld doubles2015

    • Author(s)
      Yuki Arano
    • Organizer
      Representation theory of groups, quantum groups, and operator algebras
    • Place of Presentation
      University of Copenhagen・コペンハーゲン(デンマーク)
    • Year and Date
      2015-06-01
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Unitary spherical representations of Drinfeld doubles2015

    • Author(s)
      荒野悠輝
    • Organizer
      2015 East Asian Core Doctorial Forum on Mathematics
    • Place of Presentation
      台湾国立大学(中華民国・台北)
    • Year and Date
      2015-01-22
    • Related Report
      2014 Annual Research Report
  • [Presentation] Irreducible spherical unitary representations of the Drinfeld double of SUq(3)2014

    • Author(s)
      荒野悠輝
    • Organizer
      作用素環論の最近の進展
    • Place of Presentation
      京都大学 RIMS(京都府京都市)
    • Year and Date
      2014-09-08
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] Central property (T) for quantum groups2014

    • Author(s)
      荒野悠輝
    • Organizer
      第49回関数解析研究会
    • Place of Presentation
      せせらぎ街道の宿たかお(岐阜県郡上市)
    • Year and Date
      2014-08-20
    • Related Report
      2014 Annual Research Report

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Published: 2015-01-22   Modified: 2024-03-26  

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