1995 Fiscal Year Final Research Report Summary
Structure and symmetries of integrable systems
| Project/Area Number |
05402001
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| Research Category |
Grant-in-Aid for General Scientific Research (A)
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| Allocation Type | Single-year Grants |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | KYOTO UNIVERSITY |
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Principal Investigator |
JIMBO Michio Kyoto Univ.Graduate School of Science Professor, 大学院理学研究科, 教授 (80109082)
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| Co-Investigator(Kenkyū-buntansha) |
KONO Akira Kyoto Univ., Graduate School of Science Professor, 大学院理学研究科, 助教授 (00093237)
UENO Kenji Kyoto Univ., Graduate School of Science Professor, 大学院理学研究科, 助教授 (40011655)
UMEDA Toru Kyoto Univ., Graduate School of Science Associate Professor, 大学院理学研究科, 助教授 (00176728)
TAKEI Yoshitsugu Kyoto Univ., Research Institute for Mathematical Science Associate Professor, 数理解析研究科, 助教授 (00212019)
SHIOTA Takahiro Kyoto Univ., Graduate School of Science Associate Professor, 大学院理学研究科, 助教授 (20243008)
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| Project Period (FY) |
1993 – 1995
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| Keywords | lattice model / correlations / quantum group / matrix integral / Painleve equation / q-difference operator / moduli space / loop group |
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| Research Abstract |
The major outcome of the present research project is as follows.
1.Jimbo pushed forward the study of space of states of lattice models. Ising and RSOS type models were formulated in much the same way as for the vertex type models, and difference equations for correlations were derived. The theory was extended to spin chains with a boundary, and physical quantities were derived including vacuum states, energy and magnetization. For critical systems an integral formula for correlations was found. In representation theory, a new level 0 action was constructed on a level one module over the quantum affine algebra.
2.Shiota constructed a notrivial solution to [P,Q] =P using a matrix integral similar to Kontsevich's. Takei showed, usuing the exact WKB analysis, that the Painleve I equation can be taken as a standard form near a simple turning point. He also constructed a general solution using the multiple scale anaysis.
3.Umeda studied a new construction of a q-analog of differential operators, which appear in the Capelli identity for GLq (n) , in terms of classical q-difference operators. This opened up a connection with Gelfand-type hypergeometric equations.
4.Ueno gave a concrete construction of projective flat connections on the vector bundles of conformal blocks over the moduli space of curves. Kono studied the free loop group of a compact simple Lie group, and found a connection between the mod p cohomology of the adjoint action of G on the closed loop group, and the integral cohomology of G.
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