Research of quantum group actions on operator algebras

Research Project

Project/Area Number	18K03317
Research Category	Grant-in-Aid for Scientific Research (C)
Allocation Type	Multi-year Fund
Section	一般
Review Section	Basic Section 12010:Basic analysis-related
Research Institution	Waseda University (2020-2022) Hokkaido University (2018-2019)
Principal Investigator	Tomatsu Reiji 早稲田大学, 教育・総合科学学術院, 教授 (70447366)
Project Period (FY)	2018-04-01 – 2023-03-31
Project Status	Completed (Fiscal Year 2022)
Budget Amount *help	¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000) Fiscal Year 2020: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000) Fiscal Year 2019: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000) Fiscal Year 2018: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Keywords	von Neumann環 / C環 / 作用素環 / テンソル圏 / 量子群 / Cテンソル圏 / 従順性 / フォンノイマン環 / 作用
Outline of Final Research Achievements	My research is to classify actions of C-tensor categories on von Neumann algebras. The action here means a -preserving tensor functor from the given category into the C*-tensor category consisting of endomorphisms on a von Neumann algebras. My main result is that we can classify centrally free cocycle actions on any von Neumann algebras up to approximately inner automorphisms. This result contains classification results for amenable discrete quantum groups of Kac type which was proved before. Moreover, the main result also gives us another proof of classification of subfactors due to Popa.
Academic Significance and Societal Importance of the Research Achievements	本研究で得られた結果は作用素環論やそれに関連する分野の発展に寄与するものである．現代社会の根底の部分を支えている数学のさらなる進展が見られたことに社会的意義がある．