| Project/Area Number |
17K14168
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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 | Osaka Metropolitan University (2022-2023)
Osaka Prefecture University (2017-2021) |
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Principal Investigator |
Kimura Yoshiyuki 大阪公立大学, 国際基幹教育機構, 特任講師 (10637010)
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| Project Period (FY) |
2017-04-01 – 2024-03-31
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| Project Status |
Completed (Fiscal Year 2023)
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| Budget Amount *help |
¥2,990,000 (Direct Cost: ¥2,300,000、Indirect Cost: ¥690,000)
Fiscal Year 2020: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2019: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2018: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2017: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
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| Keywords | 代数学 / 表現論 / 量子座標環 / 双対標準基底 / 量子クラスター代数 / クラスター代数 / 標準基底 |
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| Outline of Final Research Achievements |
In this research, in a joint work with Hironori Oya, we constructed the quantum analogue of twist automorphisms for quantum unipotent cells and demonstrated that they preserve the dual canonical bases. This achievement resolved the Berenstein-Rupel conjecture. Additionally, utilizing the additive categorification by Geiss, Leclerc, and Schroer, we proved that quantum cluster monomials are preserved by these quantum twist automorphisms. Furthermore, in a joint work with Fan Qin and Qiaoling Wei, we formulated twist automorphisms in quantum cluster algebras and showed that quantum cluster monomials are preserved.
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| Academic Significance and Societal Importance of the Research Achievements |
量子クラスター代数は、数学と物理学の融合を促進する重要な研究分野です。多元環の表現論、高階タイヒミュラー理論、トーリック退化、Donaldson-Thomas不変量、Poisson Lie群、離散可積分系など多岐にわたる分野と関連し、新しい代数構造の理解を深めます。特に、量子群の表現論における標準基底との関係を明らかにすることで、理論の発展に貢献します。
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