1990 Fiscal Year Final Research Report Summary
Studies of Algebraic Analysis
| Project/Area Number |
63460005
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| Research Category |
Grant-in-Aid for General Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Research Field |
解析学
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| Research Institution | Kyoto University |
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Principal Investigator |
KASHIWARA Masaki Kyoto University. Research Institute for Mathematical Sciences, Professor, 数理解析研究所, 教授 (60027381)
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| Co-Investigator(Kenkyū-buntansha) |
NAKANISHI Noboru Kyoto University. Research Institute for Mathematical Sciences, Professor, 数理解析研究所, 教授 (30027362)
TAKASAKI Kanehisa Kyoto University. Research Institute for Mathematical Sciences, Instructor, 数理解析研究所, 助手 (40171433)
SAITO Morihiko Kyoto University. Research Institute for Mathematical Sciences, Instructor, 数理解析研究所, 助手 (10186968)
MIWA Tetsuji Kyoto University. Research Institute for Mathematical Sciences, Associate Profes, 数理解析研究所, 助教授 (10027386)
KAWAI Takahiro Kyoto University. Research Institute of Mathematical Sciences, Professor, 数理解析研究所, 教授 (20027379)
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| Project Period (FY) |
1988 – 1990
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| Keywords | Kac-Moody algebra / Quantum group / Solvable lattice model / R matrix / Chiral Potts model / Hodge theory / unnarmonic oscillator / quantum gravity |
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| Research Abstract |
Kashiwara proved the generalized Kazhdan-Lustzig conjecture for Kac-Moody Lie algebras. He also proved the existence of special basis with certain nice properties at q=0 for representations of quantum groups. He named it crystal base and developed the theory. Miwa showed that crystal base is equivalent to the paths appearing in the computation of 1 point function of solvable lattice models, and, by using this fact, constructed the crystal base for the affine Lie algebras of type A. He also studied the representation theory of quantum when q is a root of unity and the related lattice models in the collaboration with Jimbo, Date, Miki. When q is a root of unity, quantum groups have a large center. The representations are classified by the values of these central elements. The R matrices exist only if they are chosen appropriately and Riimann surfaces appear form this condition. The corresponding lattice models, Which are the generalized chiral potts models, Were constructed. Morihiko Saito formulated the mixed Hodge modules as characteristic 0 analogue of the mixed Hodge theory of Deligne et al. Kawai studied the WKB method in connection with the unharmonic oscillators with Aoki and Takei, and obtained the canonical form of Schrodinger equation in the case of 1 or 2 simple turning points. Takasaki also studied the WKB method connecting Olver's method and the method of Jost solution in the scattering theory, and obtained multiple integral series for the Voros coefficients. Nakanishi studied the covariant canonical quantization of quantum gravity with Abe. They succeeded to obtain a consistent 1 order perturbation in Einstein gravitational constant, Without using a c-number background metric.
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