Affine Lie algebra characters and Bethe Ansatz

仿射李代数字符和 Bethe Ansatz

基本信息

批准号：
11640027
负责人：
OKADO Masato
金额：
$ 2.37万
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：
1999
资助国家：
日本
起止时间：
1999 至 2001
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640027/
关键词：
Affine Lie algebra quantum group integrable system

项目摘要

In this research, we have investigated affine Lie algebra characters using a method in solvable lattice models, Bethe Ansatz. We also obtained important results on cellular automata which had not been predicted at the beginning of the project.1. Fermionic formula. Fermionic formula is a polynomial with positive integer coefficients arising from combinatorics of Bethe Ansatz. We conjectured that this polynomial gives the branching function of an integrable representation of an affine Lie algebra, and considered its evidence using the crystal theory in quantum group. We also proved this conjecture in several cases.2. Combinatorics of Bethe Ansatz. Besides fermionic formulae, there is an important sysmtem of algebraic equations, called Q-system, in combinatorial studies of Bethe Ansatz. Kuniba, with Nakanishi et al., obtained a solution of this Q-system from Bethe equations at q = 0.3. Soliton cellular automaton. Although this research was not in our mind at the beginning, there was a new progress by us in the studies of soliton sellular automata. A cellular automaton is defined from the crystal of a finite dimensional representation of a quantum affine algebra. We showed that the motion of solitons in this cellular automaton factorizes into the product of 2-body ones and their scattering rule is explicitly given using the combinatorial R of finite crystals.4. Discrete integrable systems. The above mentioned soliton cellular automaton in the case of affine Lie algebra An^<(1)> has been known to be obtained from the ultra discrete limit of the nonautonomous discrete KP equation. Nagai et al. proved the solitonical nature and constructed conserved quantities from this approach.

在这项研究中，我们使用可解决的晶格模型Bethe Ansatz中的方法调查了Aggrine Lie代数字符。我们还获得了在项目开始时未预测的细胞自动机的重要结果。1。费米式公式。费米式公式是由贝特·安萨兹（Bethe Ansatz）组合引起的，具有正整数系数的多项式。我们猜想该多项式赋予了仿射谎言代数的可集成表示的分支函数，并使用量子组中的晶体理论考虑了其证据。我们还证明了这种猜想在几种情况下2。 Bethe Ansatz的组合。除了费米原子公式外，在伯特·安萨兹（Bethe Ansatz）的组合研究中，还有一个重要的代数方程式系统，称为Q-System。 Kuniba与Nakanishi等人一起，从Q = 0.3的Bethe方程中获得了该Q系统的解决方案。孤子细胞自动机。尽管这项研究一开始就没有在我们的脑海中，但是在Soliton Sellular Automata的研究中，我们取得了新的进步。细胞自动机是根据量子仿射代数的有限维表示的晶体定义的。我们表明，使用有限晶体的组合r，明确给出了这种细胞自动机中孤子的运动将其分解为2体散射规则。4。离散的集成系统。已知在非自主离散KP方程的超离散限制中获得上述孤子细胞自动机。 Nagai等。通过这种方法证明了独立性和构造的保守数量。