2021 Fiscal Year Final Research Report
Study of new algebraic and combinatorial structure in integrable systems
| Project/Area Number |
16H03922
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Nagoya University |
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Principal Investigator |
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| Co-Investigator(Kenkyū-buntansha) |
国場 敦夫 東京大学, 大学院総合文化研究科, 教授 (70211886)
尾角 正人 大阪市立大学, 大学院理学研究科, 教授 (70221843)
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| Project Period (FY) |
2016-04-01 – 2021-03-31
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| Keywords | 団代数 / 団散乱図式 / 二重対数関数 / 五角関係式 |
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| Outline of Final Research Achievements |
We study theory and applications of cluster algebras which is one of algebraic and combinatorial structure in integrable systems. We obtain the following new results and insights: derivation of dilogarithm identities in cluster algebras by classical mechanical method, systematic proofs of synchronicity phenomenon and related conjectures, derivation of the relation between cluster algebra theory and scattering diagrams in the cluster algebraic point of view, proof that any consistency relation in a cluster scattering diagram is generated by the pentagon identity among dilogarithm elements, dilogarithm identities associated to cluster scattering diagrams and their quantizations, derivation of product formula of F-polynomials.
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| Free Research Field |
団代数
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| Academic Significance and Societal Importance of the Research Achievements |
団代数は2000年頃にFominとZelevinskyにより導入された代数的組み合わせ論的構造であるが，ルート系と同様に，さまざまな数学的対象に共通して現れる基盤的な構造であることから，世界中の多くの数学者や物理学者により，その基礎と応用の研究が盛んになされている．本研究成果は団代数の基礎理論に対して，新しい多くの知見を与えるもので，今後長い期間において世界に共有される有用な科学的成果と考えている．
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