| Project/Area Number |
16K17568
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| Research Category |
Grant-in-Aid for Young Scientists (B)
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| Allocation Type | Multi-year Fund |
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| Research Field |
Algebra
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| Research Institution | Shibaura Institute of Technology |
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Principal Investigator |
Shimizu Kenichi 芝浦工業大学, システム理工学部, 准教授 (70624302)
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| Project Period (FY) |
2016-04-01 – 2020-03-31
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| Project Status |
Completed (Fiscal Year 2019)
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| Budget Amount *help |
¥2,600,000 (Direct Cost: ¥2,000,000、Indirect Cost: ¥600,000)
Fiscal Year 2019: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2018: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2017: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2016: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
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| Keywords | ホップ代数 / テンソル圏 / モジュラーテンソル圏 / 代数学 |
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| Outline of Final Research Achievements |
We extended fundamental tools in the theory of finite-dimensional Hopf algebras (integrals, Fourier transforms and characters) to the setting of finite tensor categories and then established basic theory for them. By using the action functor, we established a method for dealing with certain (co)ends in finite tensor categories. As applications, we obtained some characterizations of modularity of ribbon finite tensor categories, a projective action of the modular group on the Hochschild cohomology of a modular tensor category, an explicit expression of the relative Serre functor for a comodule algebra, and explicit expressions of the modified trace and related algebraic structures in the representation category of a quasi-Hopf algebra.
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| Academic Significance and Societal Importance of the Research Achievements |
この研究では，有限次元ホップ代数の研究における基本的な道具立てが非常に一般的な設定のもとで使えるように拡張された。数理物理学や低次元トポロジーの研究を動機として，“非半単純”なモジュラーテンソル圏の研究が活発化している。本研究の成果は，このようなテンソル圏の研究の基礎として重要であろう。モジュラー性の特徴づけ，相対セール関手と関連する代数的構造，修正トレースに関する結果などは，テンソル圏や関連する分野における今後の応用が期待される。
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