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，特别是离散量子系统中可积系统与Yang-Baxter方程椭圆解的关系。特别地，我们集中研究了本课题的主要问题，即澄清Sklyanin等人关于Calogero型系统的研究中出现的类量子群结构和离散时间演化结构。结果如下。Hasegawa得到了离散Painleve系统中Weyl群作用的一种量化，并推广了Noumi和Yamada所研究的Weyl群作用。Yang-Baxter方程的解即手性-波茨模型的解是Noumi - Yamada - Kajiwara正则变换或离散时间演化的量子生成函数。我们注意到，在这种量子情况下，变换的互易性失效，并产生辫状群作用。这些发现以一种意想不到的方式将手性波茨模型和Painleve方程联系起来，这样相互关系将有助于澄清这些物体的性质。另一方面，众所周知，Panlev型方程可以通过KP层次的约简得到。Kuroki试图扩展Hasegawa通过泊松结构量化Paileve系统的结果。他还研究了椭圆曲线上一类特殊的共形块，以及可积系统的高维推广。

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.