STUDY OF DISCRETE TIME EVOLUTIONS IN DISCRETE QUANTUM INTEGRABLE SYSTEMS

本课题旨在研究数学物理中的可积系统，特别是离散化量子系统中的可积系统与Yang-Baxter方程椭圆解之间的关系。本课题的主要内容是阐明Sklyanin等人在Calogero类系统上所做的工作中出现的量子群结构和离散时间演化结构。长谷川得到了离散Painleve系统中Weyl群作用的量子化，并推广了Noumi和Yamada的研究。杨-巴克斯特方程的解称为手征-波茨模型，它起着Noumi-Yamada-Kajiwara正则变换或离散时间演化的量子母函数的作用。我们注意到，在这种量子情况下，变换的有序性是失败的，并产生辫子群作用。这些发现以一种意想不到的方式将手征Potts模型和Painleve方程联系在一起，从而有助于阐明这些对象的性质。另一方面，众所周知，Panlev‘e型方程可以通过降低Kp阶数来得到。黑木试图推广长谷川通过泊松结构量子化帕利夫系统的结果。他还研究了椭圆曲线上的一种特殊类型的共形块，以及可积系的高维推广。

The aim of this project is to study the integrable systems in mathematical physics, especially the relationship between those systems in discretized quantum system and the elliptic solutions of the Yang-Baxter equations.In particular, we have been concentrated on the study related to the main isuue in this subject which is to clarify the quantum group like structures and the discrete time evolution structures appeared in Sklyanin and others' work on Calogero type systems.The result is as follows. Hasegawa obtained a quantization of the Weyl group action in discrete Painleve system and the generalization thereof studied by Noumi and Yamada. The solution of the Yang-Baxter equation known as the Chiral-Potts model plays the role of quantum generating function of Noumi - Yamada - Kajiwara's canonical transformation or the discrete time evolution. We remark that the invotutivity of the transformations fails in this quantum case and generate a braid group action.These findings connect the Chiral Potts models and the Painleve equations in an unexpected way so that the mutual relationship will help to clarify the nature of these objects.On the other hand, it is well known that Panlev'e type equations can be obtained by reductions of the KP hierachy. Kuroki tried to extend Hasegawa's result of quantizing the Paileve systems via the Poisson structures. Also he studied the conformal block of a special kind over the elliptic curves, as well as studied a higher dimensional generalization of the integrable systems.