Discretization and quantization of integrable and isomonodromic systems

本研究的目的之一是量子化由Berenstein-Kazhdan几何晶体产生的离散经典动力学系统。Kajiwara，Noumi和Yamada(2001)对任意正整数m和n，构造了(m，n)-矩阵空间上扩展的A型仿射Weyl群的直积的二次作用量，这是几何晶体离散经典动力系统的一个重要例子。利用A型仿射量子群，黑木构造了相互质数m和n的双子作用量的量子化。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，他还澄清了量子群和q-差分二元Weyl群作用(q-差分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，q-差量子双星Weyl群作用I…更多的S是由量子泛包络代数中下Chvalley生成元的复幂生成的，这种构造重现了长谷川构造的q-差量子双粒子作用量。因此，我们可以用量子群的语言来理解q-差量子Painleve系统。他还指出了量子L算符或以albl=lcld关系为特征的量子群的重要性。通过FRT结构，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关系来处理具有二元Weyl群作用的量子系统。他猜想q-差量子二元Weyl群作用的量子不变多项式是由由某种ALBL=LCLD关系产生的相互交换的转移矩阵产生的。他在名古屋大学2007年3月5-8日的国际研讨会上公布了上述大部分结果。长谷川(在他的预印本2007)中构造了，对于任何可对称化的GCM，由截断的q-Serre关系刻画的代数上的q-差量子双星Weyl群作用，量子化了Panleve VI系统.Kikuchi证明了常微分Painleve VI系统和q-差Painleve VI系统可以分别用某些微分和q-差孤子系统的相似约化来识别.较少

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