Tensor categories and subfactors

2022 Fiscal Year Final Research Report

• Project/Area Number: 18K13424
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 12010:Basic analysis-related
• Research Institution: Kyoto University
• Principal Investigator: Arano Yuki 京都大学, 理学研究科, 助教 (40805222)
• Project Period (FY): 2018-04-01 – 2023-03-31
• Keywords: 作用素環 / 量子群 / テンソル圏

Outline of Final Research Achievements

The theory of subfactors can be regarded as an analogue of the Galois theory in the operator algebra. It played an important role in the knot theory, the theory tensor categories and the conformal field theory. Such subfactor theory can be interpreted in terms of the actions of tensor categories on operator algebras. In this study, I imported this viewpoint to the C*-algebras, which is a different kind of operator algebra from the factors and studied the actions of tensor categories on C*-algebras toward the classification.
Especially, I classified the Rokhlin actions and formulated the equivariant KK-theory for tensor category actions, which is a homotopy theoretical aspect of the actions.

Free Research Field

作用素環論

Academic Significance and Societal Importance of the Research Achievements

作用素環論の研究において、その分類は中心的な問題として様々な研究を牽引してきた。
本研究においては、部分因子環論のC*-環におけるアナロジーを考えることにより、C*-環へのテンソル圏作用についての理解を深めた。これはC*-環の分類理論を推し進めるだけでなく、テンソル圏そのものの研究や場の量子論などの数理物理にも貢献するものである。