2006 Fiscal Year Final Research Report Summary
Discretization and quantization of integrable and isomonodromic systems
| Project/Area Number |
17540185
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Global analysis
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| Research Institution | Tohoku University |
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Principal Investigator |
KUROKI Gen Tohoku University, Graduate School of Science, Research Assistant, 大学院理学研究科, 助手 (10234593)
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| Co-Investigator(Kenkyū-buntansha) |
HASEGAWA Koji Tohoku University, Graduate School of Science, Lecturer, 大学院理学研究科, 講師 (30208483)
KIKUCHI Tetsuya Tokyo University, Graduate School of Mathematical Sciences, COE fellow, 大学院数理科学研究科, 研究拠点形成特任研究員 (00374900)
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| Project Period (FY) |
2005 – 2006
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| Keywords | integrable systems / isomonodromic systems / Painleve systems / discretization / quantization / conformal field theory / quantum group |
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| Research Abstract |
One of the aims of this research is to quantize discrete classical dynamical systems arinsing from Berenstein-Kazhdan geometric crystals. Kajiwara, Noumi, and Yamada (2001) constructed, for any positive integers m and n, the birational action of the direct product of the extended affine Weyl groups of A-type on the space of the (m, n)-matrices, which is an important example of the discrete classical dynamical system arising from a geometric crystal. Using the affine quantum groups of A-type, Kuroki has constructed, for mutually prime m and n, the quantization of the birational action. This result would be the first step for understanding the relationship between quantum groups and geometric crystals.
Furthermore, as a byproduct, he clarified the relationship between quantum groups and q-difference birational Weyl group actions (q-difference Painleve systems). He has shown that, for any symmetrizable generalized Cartan matrix (GCM), the q-difference quantum birational Weyl group action i
… More
s generated by the complex powers of the lower Chevalley generators in the quantum universal enveloping algebra and this construction reproduces the q-difference quantum birational actions constructed by Hasegawa. Thus we can understand q-difference quantum Painleve systems in the language of quantum groups.
He also has pointed out the importance of the quantum L-operators or quantum groups characterized by the ALBL=LCLD relations. By the FRT construction, quantum groups can be characterized by the RLL=LLR relations. We need, however, the more general ALBL=LCLD relations to deal with quantum systems with birational Weyl group actions. He conjectured that quantum invariant polynomials of the q-difference quantum birational Weyl group action are generated by the mutually commuting transfer matrices arising from a certain ALBL=LCLD relation.
He announced most of the results mentioned above in the international workshop "Exploration of New Structures and Natural Constructions in Mathematical Physics" at Nagoya University, 5-8 March 2007
Hasegawa (in his preprint 2007) has constructed, for any symmetrizable GCM, a q-difference quantum birational Weyl group action on the algebra characterized by truncated q-Serre relations and has quantized the Panleve VI system.
Kikuchi has shown that ordinary differential Painlve VI sysmte and the q-difference Painleve VI system can be identified with the similarity reductions of certain differential and q-difference soliton systems respectively. Less
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Research Products
(6 results)
