Discretization and quantization of integrable and isomonodromic systems

本文的目的之一是研究Berenstein-Kazhdan几何晶体中离散的经典动力学系统。Kajiwara，Noumi，and Yamada（2001）对任意正整数m和n构造了A-型扩张仿射Weyl群在（m，n）-矩阵空间上的直积的双有理作用，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。利用A型仿射量子群，Kuroki构造了互素m和n的双有理作用量的量子化。这一结果将是理解量子群与几何晶体之间关系的第一步，而且，作为副产品，他澄清了量子群与q-差双有理Weyl群作用（q-差Painleve系统）之间的关系。他证明了，对于任何可对称化的广义Cartan矩阵（GCM），q-差量子双有理Weyl群作用量是 ...更多信息 s是由量子泛包络代数中的下Chevalley生成元的复幂生成的，这种构造再现了Hasegawa构造的q-差量子双有理作用量.他还指出了用ALBL=LCLD关系刻画的量子L-算符或量子群的重要性。通过FRT构造，量子群可以用RLL=LLR关系来刻画。然而，我们需要更一般的ALBL=LCLD关系来处理具有双有理Weyl群作用的量子系统。他在名古屋大学的“数学物理中新结构和自然构造的探索”国际研讨会上宣布了上述大部分结果。2007年3月5日至8日（在他2007年的预印本中）为任何可对称化的GCM构造了，本文给出了一个q-差量子双有理Weyl群作用在以截断q-Serre关系为特征的代数上，并量子化了Panleve VI系统.菊池证明了常微分Painlve VI系统和q-差量子双有理Weyl群作用在Q-Serre关系上的量子化差分Painleve VI系统可以分别用某些差分孤子系统和q-差分孤子系统的相似约化来识别。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