2023 Fiscal Year Final Research Report
Application of cluster algebras to punctured Riemann surfaces and combinatorial representation theory
| Project/Area Number |
19K03440
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Review Section |
Basic Section 11010:Algebra-related
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| Research Institution | Chiba University |
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Principal Investigator |
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| Project Period (FY) |
2019-04-01 – 2024-03-31
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| Keywords | クラスター代数 / タイヒミュラー空間 / ワイル群 / 量子群 / q指標 / 四面体方程式 |
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| Outline of Final Research Achievements |
We studied applications of cluster algebras to representation theory and integrable systems. For a finite dimensional simple Lie algebra g, we define m-periodic quiver and realize the Weyl group of g as mutation sequences of the quiver. In particular, when q is a root of unity we clarify the Weyl invariant subgroup of a rational functional field including the q-character of the qunatum group for g. In 2020 and 2021, our research activities were severely restricted due to the pandemic, but from 2022 onwards, our research activities have gradually recovered, and we gave a large number of research talks. In 2023, after extending the research period, we started the new challenge of applying cluster algebra to three-dimensional integrable systems, and achieved great results.
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| Free Research Field |
数理物理学、可積分系
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| Academic Significance and Societal Importance of the Research Achievements |
クラスター代数の表現論、3次元可積分系への新しい応用を見つけたことが本研究の大きな学術的意義である。本研究で構成したワイル群のクラスター代数による実現は、表現論だけでなくタイヒミュラー空間や可積分系でも様々な応用が見つかっている。このワイル群の実現はアフィンLie環の場合に拡張することができ、さらなる発展が期待される。3次元可積分系へのクラスター代数の応用は、これまで発見的に構成されていた四面体方程式と3次元方程式の様々な解を統一的に扱う可能性をもった新しい手法である。
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