Integral development of theory of operator algebras

• Project/Area Number: 19H00640
• Research Category: Grant-in-Aid for Scientific Research (A)
• Allocation Type: Single-year Grants
• Section: 一般
• Review Section: Medium-sized Section 12:Analysis, applied mathematics, and related fields
• Research Institution: The University of Tokyo
• Principal Investigator: 河東 泰之 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 2022)
• Budget Amount: ¥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 / 部分因子環 / 共形場理論 / 頂点作用素代数 / テンソル圏 / フュージョン圏 / 組み紐圏 / 双ユニタリ接続 / 交換4角形 / 部分因子環論 / トポロジカル相 / 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 Annual Research Achievements

作用素環論の立場からテンソル圏の研究を行っている．テンソル圏における対象を作用素環論の立場からとらえる方法は3つある．III型因子環の自己準同型，(II型の)因子環上の bimodule, bi-unitary connection である．前の2つについては多くの研究成果があるが，3つ目のものは，1990年代には基礎的なことが知られていたにもかかわらず，あまり研究が進んでいない．
その中で，α誘導と呼ばれる表現論的手法がある．これは，抽象的には組み紐圏上の Frobenius algebra（作用素環の枠組みでは Q-system とも呼ばれる）から，誘導表現の手法で新しいフュージョン圏を作り出すテンソル関手である．もともとはカイラル共形場理論についての誘導表現として Longo-Rehren によって導入され，Bockenhauer-Evans-Kawahigashi によってさらに一般的な状況で研究され，現在ではよく知られた標準的な手法となっている．
このα誘導は当初，III型因子環の自己準同型についてもものとして研究され，またII型の因子環上の bimodule についてもよく理解されている．後者のものは抽象的な組み紐圏での理解ともごく近い．今回はこのα誘導について bi-unitary connection の立場からの記述を行い，古い Ocneanu による Dynkin 図形上の多くの例がこの枠組みで統一的に理解できることも明らかにした．この bi-unitary connection の手法は，物性物理学に現れるある種の 4-tensor と数学的に同じものなので，この方面への応用が期待される．

Research Progress Status

令和4年度が最終年度であるため、記入しない。

Strategy for Future Research Activity

令和4年度が最終年度であるため、記入しない。

Research Products

• [Journal Article] $\alpha$-induction for bi-unitary connections (2024)
  • Author(s): Kawahigashi Yasuyuki
  • Journal Title: Quantum Topology Volume: - Issue: 3 Pages: 503-536
  • DOI: 10.4171/qt/209
  • Peer Reviewed
• [Journal Article] Subfactors and mathematical physics (2023)
  • 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
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] A characterization of a finite-dimensional commuting square producing a subfactor of finite depth (2023)
  • Author(s): Y. Kawahigashi
  • Journal Title: Internat. Math. Res. Notices Volume: 2023 Issue: 10 Pages: 8419-8433
  • DOI: 10.1093/imrn/rnac082
  • Peer Reviewed
• [Journal Article] Subfactors and mathematical physics (2023)
  • Author(s): D. E. Evans and Y. Kawahigashi
  • Journal Title: Bull. Amer. Math. Soc. Volume: -
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Projector matrix product operators, anyons and higher relative commutants of subfactors (2023)
  • Author(s): Yasuyuki Kawahigashi
  • Journal Title: Math. Ann, Volume: - Issue: 3-4 Pages: 2157-2172
  • DOI: 10.1007/s00208-022-02519-0
  • Peer Reviewed / Open Access
• [Journal Article] Two-dimensional topological order and operator algebras (2021)
  • Author(s): Y. Kawahigashi
  • Journal Title: Internat. J. Modern Phys. B Volume: - Issue: 08 Pages: 2130003-2130003
  • DOI: 10.1142/s0217979221300036
• [Presentation] Quantum symmetries in operator algebras and mathematical physics (2024)
  • Author(s): Y. Kawahigashi
  • Organizer: East Asian Core Doctoral Forum on Mathematics
  • Int'l Joint Research / Invited
• [Presentation] Tensor networks, two-dimensional topological order and operator algebras (2023)
  • Author(s): Y. Kawahigashi
  • Organizer: Emerging Platforms for Quantum Computing
  • Int'l Joint Research / Invited
• [Presentation] α-induction for bi-unitary connections (2023)
  • Author(s): Y. Kawahigashi
  • Organizer: Where Mathematics Meets Quantum Physics: a workshop on the occasion of Roberto Longo's 70th birthday
  • Int'l Joint Research / Invited
• [Presentation] α-induction for bi-unitary connections (2023)
  • Author(s): Y. Kawahigashi
  • Organizer: OAS Follow on: Operator Algebras: Subfactors and Applications
  • Int'l Joint Research / Invited
• [Presentation] Operator algebras, tensor categories and quantum field theory (2023)
  • Author(s): Y. Kawahigashi
  • Organizer: Workshop on Operator Algebras, Deformation Quantization and Related Field Theories
  • Int'l Joint Research / Invited
• [Presentation] Quantum symmetries in operator algebras and mathematical physics (2023)
  • Author(s): Y. Kawahigashi
  • Organizer: International Congress on Basic Science
  • Int'l Joint Research / Invited
• [Presentation] Two-dimensional topological order and operator algebras (2023)
  • Author(s): Y. Kawahigashi
  • Organizer: Topological Quantum Computation, International Centre for Mathematical Sciences
  • Int'l Joint Research / Invited
• [Presentation] Bi-unitary connections, modular tensor categories and alpha-induction (2023)
  • Author(s): Y. Kawahigashi
  • Organizer: Subfactors and Fusion (2-)Categorie
  • Int'l Joint Research / Invited
• [Presentation] A characterization of a finite-dimensional commuting square producing a subfactor of finite depth (2022)
  • Author(s): Y. Kawahigashi
  • Organizer: Topological Quantum Groups, C*-Tensor Categories, and Subfactors
  • Int'l Joint Research / Invited
• [Presentation] A characterization of a finite-dimensional commuting square producing a subfactor of finite depth (2022)
  • Author(s): Y. Kawahigashi
  • Organizer: New Frontiers: Interactions between Quantum Physics and Mathematics
  • Int'l Joint Research / Invited
• [Presentation] A characterization of a finite-dimensional commuting square producing a subfactor of finite depth (2022)
  • Author(s): Y. Kawahigashi
  • Organizer: Operator Theory 28
  • Int'l Joint Research / Invited
• [Presentation] A characterization of a finite-dimensional commuting square producing a subfactor of finite depth (2022)
  • Author(s): Y. Kawahigashi
  • Organizer: Operator Algebras: Subfactors, K-theory, Conformal Field Theory
  • Int'l Joint Research / Invited
• [Presentation] Mathematics of topological phases of matter and operator algebras (2022)
  • Author(s): Y. Kawahigashi
  • Organizer: Pan-Pacific Workshop on Topology and Correlation in Exotic Materials
  • Int'l Joint Research / Invited
• [Presentation] Tensor networks and operator algebras (2022)
  • Author(s): Y. Kawahigashi
  • Organizer: Complex geometry and related topics
  • Int'l Joint Research / Invited
• [Presentation] Tensor networks, commuting squares and higher relative commutants of subfactors (2021)
  • Author(s): Y. Kawahigashi
  • Organizer: Special Week on Operator Algebras 2021
  • Int'l Joint Research / Invited
• [Presentation] Tensor networks, commuting squares and higher relative commutants of subfactors (2021)
  • Author(s): Y. Kawahigashi
  • Organizer: Special Session: Quantum groups and algebraic quantum field theory, International Workshop on Operator Theory and its Applications
  • Int'l Joint Research / Invited
• [Presentation] Topological order, tensor categories and operator algebras (2021)
  • Author(s): Y. Kawahigashi
  • Organizer: MSJ-KMS Joint Meeting 2021
  • Int'l Joint Research / Invited
• [Presentation] Tensor networks, commuting squares and higher relative commutants of subfactors (2021)
  • Author(s): Y. Kawahigashi
  • Organizer: Subfactors, Vertex Operator Algebras, and Tensor Categories
  • Int'l Joint Research / Invited
• [Presentation] Topological order, tensor networks and operator algebras (2021)
  • Author(s): Y. Kawahigashi
  • Organizer: Quantum Math, Singularities and Applications
  • Int'l Joint Research / Invited
• [Presentation] Connections, subfactors and tensor networks (2021)
  • Author(s): Y. Kawahigashi
  • Organizer: Fusion categories and tensor networks
  • Int'l Joint Research / Invited
• [Presentation] Topological order, tensor networks and subfactors (2020)
  • Author(s): Y. Kawahigashi
  • Organizer: Topological Orders and Higher Structures
  • Int'l Joint Research / Invited
• [Remarks] 河東泰之
  • URL: https://www.ms.u-tokyo.ac.jp/~yasuyuki/
• [Remarks] 河東泰之のページ
  • URL: https://www.ms.u-tokyo.ac.jp/~yasuyuki/