| Project/Area Number |
17K18728
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| Research Category |
Grant-in-Aid for Challenging Research (Exploratory)
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| Allocation Type | Multi-year Fund |
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| Research Field |
Algebra, Geometry, and related fields
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| Research Institution | Waseda University |
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Principal Investigator |
Murakami Jun 早稲田大学, 理工学術院, 教授 (90157751)
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| Project Period (FY) |
2017-06-30 – 2021-03-31
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| Project Status |
Completed (Fiscal Year 2020)
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| Budget Amount *help |
¥5,980,000 (Direct Cost: ¥4,600,000、Indirect Cost: ¥1,380,000)
Fiscal Year 2019: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
Fiscal Year 2018: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
Fiscal Year 2017: ¥2,340,000 (Direct Cost: ¥1,800,000、Indirect Cost: ¥540,000)
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| Keywords | 結び目 / 量子不変量 / 量子群 / 双曲幾何 / ３次元多様体 / ホップ代数 / 基本群 / SL(2) 表現 / 量子化 / 表現論 / ジョーンズ多項式 / 体積予想 / 表現 / 双曲構造 / 結び目理論 |
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| Outline of Final Research Achievements |
As the first step of a quantization of the geometric structure of 3-manifolds, it is tried to quantize the SL(2, C) representation of the fundamental group of a knot complement. For braided Hopf algebras, including the braided quantum group BSL(2, C), a representation space is constructed from the knot. The isomorphism class of such representation spaces is proved to be a knot invariant. Especially, the representation space for BSL(2, C) can be considered as the quantization of the representation space for SL(2, C) representation.
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| Academic Significance and Societal Importance of the Research Achievements |
幾何学においては、量子化の考え方は、変形理論や非可換幾何学の構成の大きな動機となっている．これまでは、幾何的な量子化は偶数次元でしか構成できないと考えられてきたが、本研究では３次元多様体の量子化を目指し、その第一歩として基本群の SL(2, C) 表現の量子化の構成に成功した．この理論をもとに、奇数次元の多様体の量子化への道が開けるものt期待している．
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