权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Combinatorial Study of Crystal Bases and its Application to Discrete Integrable Systems

晶体基的组合研究及其在离散可积系统中的应用

基本信息

批准号：
14540026
负责人：
OKADO Masato
金额：
$ 2.56万
依托单位：
OSAKA UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2002
资助国家：
日本
起止时间：
2002 至 2004
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14540026/
关键词：
Integrable system Quantum group Yang-Baxter equation Cell automaton

项目摘要

(i) Geometric crystalGeometric crystal, indtoduced axiomatically by Berenstein and Kazhdan, was consttructed explicitly for type D^(1)_n of Kac-Moody Lie algebras. By using a matrix realization of this geometric crystal, we constructed a birational mapping on the product (tropical R) commuting with the action of the geometric crystal and also,showed that it satisfies the Yang-Baxter equation. It is believed that a geometric crystal exists arrogated to each vertex of the Dynkin diagram corresponding to the Lie algebra. For type D, there are n vertices, so it is expected that there are n distinct geometric crystals. During the last yeah we calculated the geometric crystal for k=2 by Mathematics in collaboration with Masaki Kashiwara at RIMS, Kyoto University. Data is huge and no way to print it out. However to write it down in a meaningful manner is, besides with the extension to the case when k is greater than 2, becomes a future problem.(ii)Crystal and soliton cellular automaton associated to an exceptional acne Lie algebraCoordinate representation of a series of finite crystals for an exceptional affine Lie algebra D_4^(3) is given and the zero action is explicitly obtained. Moreover we constructed a cell automaton corresponding to this series of crystals and determined the internal degree of freedom of the solitons appearing in the system and the scattering rule of two solitons.(iii)Box-ball system with reflecting endWe have extended the box-ball system, an important example of ultra discrete integrable systems, to the case with one reflecting end. Similar to the usual box-ball system, it has an infinite family of commuting time evolutions and conserved quantities associated to each time evolution. We also defined soliton states and described the reflection rule of one soliton and the scattering rule of two solitons in terms of combinatorics of crystals.

(i)几何晶体Berenstein和Kazhdan公理化地引入了几何晶体，并对Kac-Moody李代数的D^（1）_n型明确地构造了几何晶体.利用这种几何晶体的矩阵实现，构造了与几何晶体作用量可交换的乘积（热带R）上的一个双有理映射，并证明了它满足Yang-Baxter方程.据信，存在一个几何晶体，其被分配到对应于李代数的Dynkin图的每个顶点。对于D型，有n个顶点，因此预期有n个不同的几何晶体。在过去的是，我们计算了几何晶体k=2的数学与Masaki Kashiwara在RIMS，京都大学合作。数据很大，无法打印出来。然而，以一种有意义的方式写下它，除了扩展到k大于2的情况之外，成为未来的问题。（ii）与例外仿射李代数相关联的晶体和孤子元胞自动机给出了例外仿射李代数D_4^（3）的一系列有限晶体的坐标表示，并明确地得到了零作用量。构造了相应的元胞自动机，确定了系统中孤子的内部自由度和两个孤子的散射规律。（iii）具有反射端的Box-ball系统我们将超离散可积系统的一个重要例子Box-ball系统推广到具有一个反射端的情形。类似于通常的盒球系统，它有一个无限族的交换时间演化和守恒量与每个时间演化。定义了孤子态，用晶体的组合学描述了一个孤子的反射规律和两个孤子的散射规律。