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) 几何晶体几何晶体由Berenstein 和Kazhdan 公理化地引入，是为Kac-Moody 李代数的D^(1)_n 型明确构造的。通过使用这种几何晶体的矩阵实现，我们构建了在几何晶体作用下的乘积（热带R）上的双有理映射，并证明它满足Yang-Baxter方程。人们相信，对应于李代数的 Dynkin 图的每个顶点都存在一个几何晶体。对于 D 型，有 n 个顶点，因此预计有 n 个不同的几何晶体。最后，我们与京都大学 RIMS 的 Masaki Kashiwara 合作，通过数学计算了 k=2 的几何晶体。数据量很大，没有办法打印出来。然而，以有意义的方式将其写下来，除了扩展到 k 大于 2 的情况之外，还成为未来的问题。 (ii) 与异常痤疮李代数相关的晶体和孤子元胞自动机一系列给出了特殊仿射李代数 D_4^(3) 的有限晶体，并显式获得了零作用。此外，我们还构建了与这一系列晶体相对应的元胞自动机，并确定了系统中出现的孤子的内部自由度以及两个孤子的散射规则。 (iii)带反射端的盒球系统我们对盒球系统进行了扩展。系统是超离散可积系统的一个重要例子，适用于具有一个反射端的情况。与通常的盒子球系统类似，它具有无限的通勤时间演化族以及与每个时间演化相关的守恒量。我们还定义了孤子态，并用晶体组合学描述了一个孤子的反射规则和两个孤子的散射规则。