| Project/Area Number |
10440007
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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 | Osaka University |
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Principal Investigator |
MURUKAMI Jun Graduate School of Science, Osaka University Associate Professor, 大学院・理学研究科, 助教授 (90157751)
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| Co-Investigator(Kenkyū-buntansha) |
UNO Katsuhiro Graduate School of Science, Osaka University Associate Prof., 大学院・理学研究科, 助教授 (70176717)
NAGATOMO Kiyokazu Graduate School of Science, Osaka University Associate Prof., 大学院・理学研究科, 助教授 (90172543)
KAWANAKA Noriaki Graduate School of Science, Osaka University Professor, 大学院・理学研究科, 教授 (10028219)
MIKI Kei Graduate School of Science, Osaka University Associate Prof., 大学院・理学研究科, 助教授 (40212229)
SAKUMA Makoto Graduate School of Science, Osaka University Associate Prof., 大学院・理学研究科, 助教授 (30178602)
山田 修司 京都産業大学, 理学部, 助教授 (30192404)
山根 宏之 大阪大学, 大学院・理学研究科, 講師 (10230517)
今野 一宏 大阪大学, 大学院・理学研究科, 助教授 (10186869)
落合 豊行 奈良女子大学, 大学院人間文化研究科, 教授 (70016179)
和久井 道久 大阪大学, 大学院・理学研究科, 助手 (60252574)
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| Project Period (FY) |
1998 – 2000
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| Project Status |
Completed (Fiscal Year 2000)
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| Budget Amount *help |
¥7,300,000 (Direct Cost: ¥7,300,000)
Fiscal Year 2000: ¥2,500,000 (Direct Cost: ¥2,500,000)
Fiscal Year 1999: ¥1,900,000 (Direct Cost: ¥1,900,000)
Fiscal Year 1998: ¥2,900,000 (Direct Cost: ¥2,900,000)
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| Keywords | Riemann surface / Knot theory / 3-dimensional manifold / representation theory / vertey operator algebra / quantum invariant / finite type invariant / 低次元位相幾何 / 3次元多様体 / 共形場理論 / 双曲構造 / 結び目 / 双曲幾何 / 有限群 / 写像類群 / ウェブ図 / 量子摂動不変量 |
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| Research Abstract |
The aim of this research is to construct a new unifying method to study low dimensional topology from a view point of the quantum structure of Riemann surfaces. To do this, research was done for two areas. One is geometric aspect and the other is algebraic aspect. It is revealed that the volume of the complement of a hyperbolic knot is given by quantum invariants of the knot. Moreover, there is some indication that the volume of a hyperbolic three-manifold is also given by the quantum invariants of the manifold. These facts suggest that the quantum invariants of knots and three-manifolds include various geometric information, and efficiency of the method to study geometric properties from quantum invariants are pointed out.
We also study about the finite-type invariants and the web diagrams, which are related to certain expansion of quantum invariants, and get some new properties of them.
Quantum invariants are closely related to conformal field theory and theory of q-deformation, and some results for these theories are obtained. In the study of vertex operator algebras, the relation between modular forms and the correlation functions in conformal field theory is given. In the study of the theory of q-deformation, modular representations of finite groups are studied. The heart (quotient of the radical by the socle) of projective indecomposable modules are investigated, and the case that the heart is not indecomposable is determined. Moreover, q-deformation of a Frobenius-Schur character of complex reflection groups is defined and computed actually for the symmetric groups and imprimitive complex reflection groups.
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