Discretization and quantization of integrable and isomonodromic systems

本研究的目的之一是量化由Berenstein-Kazhdan几何晶体产生的离散经典动力系统。Kajiwara, Noumi, and Yamada（2001）构造了任意正整数m和n上a型扩展仿射Weyl群的直积在(m, n)-矩阵空间上的双象作用，这是由几何晶体产生的离散经典动力系统的一个重要例子。Kuroki利用a型仿射量子群，对互素数m和互素数n，构造了双相作用的量子化。这一结果将是理解量子群和几何晶体之间关系的第一步。此外，作为副产品，他澄清了量子群与q差双态Weyl群作用（q差Painleve系统）之间的关系。他证明了，对于任何可对称的广义Cartan矩阵（GCM），由量子普适包膜代数中下Chevalley发生器的复幂产生的q-差分量子双态Weyl群作用i…More s，这种构造再现了Hasegawa构造的q-差分量子双态Weyl群作用。因此，我们可以用量子群的语言来理解q差量子痛级系统。他还指出了以ALBL= lld关系为特征的量子l算子或量子群的重要性。通过FRT构造，量子群可以用RLL=LLR关系来表征。然而，我们需要更一般的ALBL= lld关系来处理具有双民族Weyl群作用的量子系统。他推测q差量子双态Weyl群作用的量子不变多项式是由某种ALBL= lld关系产生的互交换传递矩阵产生的。他在2007年3月5日至8日于名古屋大学举行的国际研讨会“探索数学物理中的新结构和自然结构”上宣布了上述大部分结果。hasegawa（在他的2007年预印本中）在截断q-Serre关系表征的代数上构造了任何对称GCM的q差分量子双态Weyl群作用，并量子化了panlevel VI系统。Kikuchi证明了用一定的微分孤子系统和q-差分孤子系统的相似约简分别可以识别常微分孤子系统和q-差分孤子系统。少

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 2007Hasegawa (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