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Integral development of theory of operator algebras

Research Project

Project/Area Number 19H00640
Research Category

Grant-in-Aid for Scientific Research (A)

Allocation TypeSingle-year Grants
Section一般
Review Section Medium-sized Section 12:Analysis, applied mathematics, and related fields
Research InstitutionThe University of Tokyo

Principal Investigator

Kawahigashi Yasuyuki  東京大学, 大学院数理科学研究科, 教授 (90214684)

Project Period (FY) 2019-04-01 – 2023-03-31
Project Status Completed (Fiscal Year 2023)
Budget Amount *help
¥45,370,000 (Direct Cost: ¥34,900,000、Indirect Cost: ¥10,470,000)
Fiscal Year 2022: ¥10,010,000 (Direct Cost: ¥7,700,000、Indirect Cost: ¥2,310,000)
Fiscal Year 2021: ¥11,830,000 (Direct Cost: ¥9,100,000、Indirect Cost: ¥2,730,000)
Fiscal Year 2020: ¥10,010,000 (Direct Cost: ¥7,700,000、Indirect Cost: ¥2,310,000)
Fiscal Year 2019: ¥13,520,000 (Direct Cost: ¥10,400,000、Indirect Cost: ¥3,120,000)
Keywords作用素環 / テンソル圏 / フュージョン圏 / 部分因子環 / 共形場理論 / トポロジカル相 / 双ユニタリ接続 / α誘導 / 頂点作用素代数 / 組み紐圏 / 交換4角形 / 部分因子環論 / 数理物理学 / テンソルネットワーク / 作用素環論
Outline of Research at the Start

近年,本計画の参加者らの活躍により,再び日本は作用素環論研究における世界の中心国の一つとなった.国際的にみても日本の作用素環論研究体制は十分優れたものであると言える.
本計画はこの日本の勢いをさらに加速し,頂点作用素代数,離散群論,指数定理,共形場理論などとの関係を一段と深めながら,作用素環論研究の一層の進展を目指すものである.そのためフィールズ賞受賞者のJones らを協力研究者に加え,作用素環論の幅広い話題をカバーした計画で,若手研究者も養成しながら世界のトップに立つことが目的である.このため,数理物理学との関係に重心を置きながら,研究を進めていく.

Outline of Final Research Achievements

We studied relations between 4-tensors appearing in 2-dimensional topolotical order and bi-unitary connections in subfactor theory in operator algebras. In particular, we proved thatt the 4-tensors in the work of Bultinck-Mariena-Williamson-Sahinoglu-Haegeman-Verstraete are essentially the same as (not-necessarily-flat) bi-unitary connections in subfactor theory.
We clarified relations between such bi-unitary connections and generalized quantum 6j-symbols. described alpha-induction in this setting, and gave an operator algebraic description of the range of the projector matrix product operators appearing in physics literature.

Academic Significance and Societal Importance of the Research Achievements

2次元トポロジカル物性物理学における最新の研究対象と,作用素環論,特にJonesの部分因子環論において1990年代から研究されていた対象が,本質的に同じものであることが明らかになり,作用素環論の新しい応用面が大きく切り開かれた.これによって,物性物理学に現れる数学的問題の理解が深く前進した.
また,このようなトポロジカル物性物理学の視点から,作用素環論についての新しい問題意識が生まれ,作用素環論,特に部分因子環論における古い研究テーマについてこれまで見逃されていた問題がいくつもあることが明らかになり,これらについて新しい解答が得られた.

Report

(6 results)
  • 2023 Final Research Report ( PDF )
  • 2022 Annual Research Report
  • 2021 Annual Research Report
  • 2020 Annual Research Report
  • 2019 Comments on the Screening Results   Annual Research Report
  • Research Products

    (29 results)

All 2024 2023 2022 2021 2020 Other

All Journal Article (6 results) (of which Int'l Joint Research: 2 results,  Peer Reviewed: 5 results,  Open Access: 3 results) Presentation (21 results) (of which Int'l Joint Research: 21 results,  Invited: 21 results) Remarks (2 results)

  • [Journal Article] $\alpha$-induction for bi-unitary connections2024

    • Author(s)
      Kawahigashi Yasuyuki
    • Journal Title

      Quantum Topology

      Volume: - Issue: 3 Pages: 503-536

    • DOI

      10.4171/qt/209

    • Related Report
      2022 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Subfactors and mathematical physics2023

    • Author(s)
      Evans David、Kawahigashi Yasuyuki
    • Journal Title

      Bulletin of the American Mathematical Society

      Volume: 60 Issue: 4 Pages: 459-482

    • DOI

      10.1090/bull/1799

    • Related Report
      2022 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] A characterization of a finite-dimensional commuting square producing a subfactor of finite depth2023

    • Author(s)
      Y. Kawahigashi
    • Journal Title

      Internat. Math. Res. Notices

      Volume: 2023 Issue: 10 Pages: 8419-8433

    • DOI

      10.1093/imrn/rnac082

    • Related Report
      2021 Annual Research Report 2020 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Subfactors and mathematical physics2023

    • Author(s)
      D. E. Evans and Y. Kawahigashi
    • Journal Title

      Bull. Amer. Math. Soc.

      Volume: -

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Projector matrix product operators, anyons and higher relative commutants of subfactors2023

    • Author(s)
      Yasuyuki Kawahigashi
    • Journal Title

      Math. Ann,

      Volume: - Issue: 3-4 Pages: 2157-2172

    • DOI

      10.1007/s00208-022-02519-0

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Two-dimensional topological order and operator algebras2021

    • Author(s)
      Y. Kawahigashi
    • Journal Title

      Internat. J. Modern Phys. B

      Volume: - Issue: 08 Pages: 2130003-2130003

    • DOI

      10.1142/s0217979221300036

    • Related Report
      2019 Annual Research Report
  • [Presentation] Quantum symmetries in operator algebras and mathematical physics2024

    • Author(s)
      Y. Kawahigashi
    • Organizer
      East Asian Core Doctoral Forum on Mathematics
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Tensor networks, two-dimensional topological order and operator algebras2023

    • Author(s)
      Y. Kawahigashi
    • Organizer
      Emerging Platforms for Quantum Computing
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] α-induction for bi-unitary connections2023

    • Author(s)
      Y. Kawahigashi
    • Organizer
      Where Mathematics Meets Quantum Physics: a workshop on the occasion of Roberto Longo's 70th birthday
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] α-induction for bi-unitary connections2023

    • Author(s)
      Y. Kawahigashi
    • Organizer
      OAS Follow on: Operator Algebras: Subfactors and Applications
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Operator algebras, tensor categories and quantum field theory2023

    • Author(s)
      Y. Kawahigashi
    • Organizer
      Workshop on Operator Algebras, Deformation Quantization and Related Field Theories
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Quantum symmetries in operator algebras and mathematical physics2023

    • Author(s)
      Y. Kawahigashi
    • Organizer
      International Congress on Basic Science
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Two-dimensional topological order and operator algebras2023

    • Author(s)
      Y. Kawahigashi
    • Organizer
      Topological Quantum Computation, International Centre for Mathematical Sciences
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Bi-unitary connections, modular tensor categories and alpha-induction2023

    • Author(s)
      Y. Kawahigashi
    • Organizer
      Subfactors and Fusion (2-)Categorie
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] A characterization of a finite-dimensional commuting square producing a subfactor of finite depth2022

    • Author(s)
      Y. Kawahigashi
    • Organizer
      Topological Quantum Groups, C*-Tensor Categories, and Subfactors
    • Related Report
      2022 Annual Research Report 2020 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] A characterization of a finite-dimensional commuting square producing a subfactor of finite depth2022

    • Author(s)
      Y. Kawahigashi
    • Organizer
      New Frontiers: Interactions between Quantum Physics and Mathematics
    • Related Report
      2022 Annual Research Report 2020 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] A characterization of a finite-dimensional commuting square producing a subfactor of finite depth2022

    • Author(s)
      Y. Kawahigashi
    • Organizer
      Operator Theory 28
    • Related Report
      2022 Annual Research Report 2020 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] A characterization of a finite-dimensional commuting square producing a subfactor of finite depth2022

    • Author(s)
      Y. Kawahigashi
    • Organizer
      Operator Algebras: Subfactors, K-theory, Conformal Field Theory
    • Related Report
      2022 Annual Research Report 2020 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Mathematics of topological phases of matter and operator algebras2022

    • Author(s)
      Y. Kawahigashi
    • Organizer
      Pan-Pacific Workshop on Topology and Correlation in Exotic Materials
    • Related Report
      2022 Annual Research Report 2020 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Tensor networks and operator algebras2022

    • Author(s)
      Y. Kawahigashi
    • Organizer
      Complex geometry and related topics
    • Related Report
      2021 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Tensor networks, commuting squares and higher relative commutants of subfactors2021

    • Author(s)
      Y. Kawahigashi
    • Organizer
      Special Week on Operator Algebras 2021
    • Related Report
      2021 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Tensor networks, commuting squares and higher relative commutants of subfactors2021

    • Author(s)
      Y. Kawahigashi
    • Organizer
      Special Session: Quantum groups and algebraic quantum field theory, International Workshop on Operator Theory and its Applications
    • Related Report
      2021 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Topological order, tensor categories and operator algebras2021

    • Author(s)
      Y. Kawahigashi
    • Organizer
      MSJ-KMS Joint Meeting 2021
    • Related Report
      2021 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Tensor networks, commuting squares and higher relative commutants of subfactors2021

    • Author(s)
      Y. Kawahigashi
    • Organizer
      Subfactors, Vertex Operator Algebras, and Tensor Categories
    • Related Report
      2021 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Topological order, tensor networks and operator algebras2021

    • Author(s)
      Y. Kawahigashi
    • Organizer
      Quantum Math, Singularities and Applications
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Connections, subfactors and tensor networks2021

    • Author(s)
      Y. Kawahigashi
    • Organizer
      Fusion categories and tensor networks
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Topological order, tensor networks and subfactors2020

    • Author(s)
      Y. Kawahigashi
    • Organizer
      Topological Orders and Higher Structures
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Remarks] 河東泰之

    • URL

      https://www.ms.u-tokyo.ac.jp/~yasuyuki/

    • Related Report
      2022 Annual Research Report 2021 Annual Research Report
  • [Remarks] 河東泰之のページ

    • URL

      https://www.ms.u-tokyo.ac.jp/~yasuyuki/

    • Related Report
      2019 Annual Research Report

URL: 

Published: 2019-04-18   Modified: 2025-01-30  

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