Study of algebraic methods for Morita dual of finite tensor categories and related algebraic structures

2023 Fiscal Year Final Research Report

• Project/Area Number: 20K03520
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 11010:Algebra-related
• Research Institution: Shibaura Institute of Technology
• Principal Investigator: Shimizu Kenichi 芝浦工業大学, システム理工学部, 准教授 (70624302)
• Project Period (FY): 2020-04-01 – 2024-03-31
• Keywords: ホップ代数 / テンソル圏

Outline of Final Research Achievements

As joint research with Taiki Shibata (Okayama University of Science), I established basic theory of Nakayama functors for coalgebras. We introduced the Nakayama functor for a coalgebra C, which is not necessarily finite-dimensional, as an endofunctor on the category of right comodules over C and showed that it is expressed by coends as in the finite-dimensional case. We also gave relation between properties of the Nakayama functor and those of the coalgebra (including semiperfectness, quasi-co-Frobenius property, and symmetric co-Frobenius property). In addition, we gave some applications to Frobenius tensor categories. I also studied Nakayama functors and Morita theory of finite tensor categories, and obtained some formulas of the Nakayama functor and characterizations of quasi-Frobenius algebras in finite tensor categories.

Free Research Field

代数学

Academic Significance and Societal Importance of the Research Achievements

本研究課題では中山関手とテンソル圏の森田理論について研究し，テンソル圏に関する様々な基礎的な結果が得られた。テンソル圏の理論は，代数学的な見地からのみならず，低次元トポロジー，作用素環論，数理物理学などの観点からも重要である。これらの分野におけるテンソル圏の研究はフュージョン圏（有限かつ半単純なテンソル圏）に対するものが多かったが，本研究では半単純の場合に知られている多くの結果を非半単純な場合に一般化しており，ここに本研究の特色がある。近年では，フュージョン圏の理論が物質のトポロジカル相と関連して盛んに研究されており，将来的には，そのような方向性からの応用も期待される。