1996 Fiscal Year Final Research Report Summary
Algebraic analysis of infinite symmetry
| Project/Area Number |
08454006
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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 |
KASHIWARA Masaki Professor, RIMS,Kyoto U., 数理解析研究所, 教授 (60027381)
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| Co-Investigator(Kenkyū-buntansha) |
IHARA Yasutaka Professor, RIMS,Kyoto U., 数理解析研究所, 教授 (70011484)
MORI Shigefumi Professor, RIMS,Kyoto U., 数理解析研究所, 教授 (00093328)
SAITO Kyoji Professor, RIMS,Kyoto U., 数理解析研究所, 教授 (20012445)
DING Jintai Lecturer, RIMS,Kyoto U., 数理解析研究所, 講師 (50273529)
MIWA Tetsuji Professor, RIMS,Kyoto U., 数理解析研究所, 教授 (10027386)
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| Project Period (FY) |
1996
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| Keywords | algebraic analysis / infinite symmetry / solvable model / quantum group / crystal base / correlation function |
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| Research Abstract |
1. Kazhdan-Lusztig Conjecture Lusztig conjectured that the irreducible character of highest weight modules with negative level is given by the Kazhdan-Lusztig polynomials. This conjecture had been proved in the integral level case. We succeeded to prove this conjecture in the non-integral case. This leads us to prove another conjecture of Lusztig on the modular representation of Chevalley groups.
2. Study of Crystal bases We succeeded to construct the Fock representations of the quantum affine algebras starting from an arbitrary finite-dimensional representations with perfect crystal. This is obtained as an analytic continuation from |q|<< 1. Furthermore we showed that this Fock representation decomposes to the tensor product of the Boson Fock space and the irreducible highest modules of the quantum affine algebras.
These results are recently applied to the study of modular representations of spin symmetric groups by the young French mathematicians Leclerc-Thibon.
3. Study of solvable models The solution to the q-KZ equation has been known when its parameter q satisfies |q|< 1. We constructed its solution when |q|=1 and studied its properties. This solution is supposed to be the correlation function of XXZ-model in the gap-less regime and we expect its further development.
4. Study of vertex operators Vertex operator is an operator on the irreducible highest weight modules indexed by finite-dimensional representation of quantum affine algebra. We analyze the vertex operator via crystal bases and leads to a new class of representations of quantum affine algebras.
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