Discretization and quantization of integrable and isomonodromic systems
Project/Area Number  17540185 
Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Global analysis

Research Institution  Tohoku University 
Principal Investigator 
KUROKI Gen Tohoku University, Graduate School of Science, Research Assistant, 大学院理学研究科, 助手 (10234593)

CoInvestigator(Kenkyūbuntansha) 
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)

Project Period (FY) 
2005 – 2006

Project Status 
Completed(Fiscal Year 2006)

Budget Amount *help 
¥2,600,000 (Direct Cost : ¥2,600,000)
Fiscal Year 2006 : ¥1,200,000 (Direct Cost : ¥1,200,000)
Fiscal Year 2005 : ¥1,400,000 (Direct Cost : ¥1,400,000)

Keywords  integrable systems / isomonodromic systems / Painleve systems / discretization / quantization / conformal field theory / quantum group / パンルヴェ方程式 / 超離散化 / 表現論 
Research Abstract 
One of the aims of this research is to quantize discrete classical dynamical systems arinsing from BerensteinKazhdan 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 Atype 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 Atype, 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 qdifference birational Weyl group actions (qdifference Painleve systems). He has shown that, for any symmetrizable generalized Cartan matrix (GCM), the qdifference 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 qdifference quantum birational actions constructed by Hasegawa. Thus we can understand qdifference quantum Painleve systems in the language of quantum groups. He also has pointed out the importance of the quantum Loperators 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 qdifference 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, 58 March 2007 Hasegawa (in his preprint 2007) has constructed, for any symmetrizable GCM, a qdifference quantum birational Weyl group action on the algebra characterized by truncated qSerre relations and has quantized the Panleve VI system. Kikuchi has shown that ordinary differential Painlve VI sysmte and the qdifference Painleve VI system can be identified with the similarity reductions of certain differential and qdifference soliton systems respectively. Less

Report
(3results)
Research Products
(9results)