Crystal bases and their application to ultradiscrete integrable systems

当我们试图理解无限维代数的本质时，一种可能的方法是找到反映代数深层性质的自然基。这样的研究有时会与数学物理中的另一个模型产生意想不到的联系。在量子仿射代数的晶体基的情况下，相应的模型是称为箱球系统的超离散孤子模型的典型例子。箱球系统的作用变量和角度变量的集合，称为操纵构型，作为一个自然的基础，即使我们考虑诸如对称代数的对称性这样的深层性质，它也具有很好的性质。

When we try to understand the nature of an infinite dimensional algebra, one possible way is to find a natural basis which reflectsdeep properties of the algebra. Such a research sometimes brings an unexpected connection with another model in mathematical physics. In the case of the crystal bases of the quantum affine algebra the corresponding model is a typical example of the ultra-discrete soliton models called the box-ball system. The set of the action and angle variables of the box-ball systems, called the rigged configurations, serves as a natural basis which has a nice property even if we consider such deep property like the symmetry of the algebra of the symmetry.