2019 Fiscal Year Final Research Report
algebraic analysis and representation theory
| Project/Area Number |
15H03608
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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 | Kyoto University |
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Principal Investigator |
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| Project Period (FY) |
2015-04-01 – 2020-03-31
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| Keywords | 箙ヘッケ環 / 団代数 / モノイダル圏化 / アフィン量子群 / D加群 / 副解析層 |
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| Outline of Final Research Achievements |
As for algebraic analysis, I studied irregular holonomic D-modules with Andrea D'Agnolo. One of main results is that the Laplace transforms of subanalytic onstructible sheaves are determined by local data. The method is totally topological. As for representation theory, I studied the cluster algebra structure of the quantum coordinate rings by using its monoidal categorification. As a byproduct, I proved that the cluster monomials correspond to simple modules in the monoidal categorification. This result solves affirmatively the conjecture due to Hernandez-Leclerc.
The key point of the proof is that a given monoidal cluster admits successive mutations once it admits the first-step mutations.
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| Free Research Field |
代数解析
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| Academic Significance and Societal Importance of the Research Achievements |
表現論は、種々の数学的対象の対称性を研究する分野である。最近、幾何学、圏論、組み合せ論など、様々な観点からの研究が進んでいる。研究代表者は、箙ヘッケ環をもちいて量子座標環のモノイダル圏化をおこない、量子座標環の団代数としての構造の解明に新しい観点を提供した。また、いろいろなアフィン量子群の間に今まで知られていなかった関係、例えば、その既約表現とテンソル積の分解係数が同じ族に属するアフィン量子群において一致することを見いだした。
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