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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.

在本研究中，我们利用可解格模型中的Bethe Answer方法研究了仿射李代数的性质。我们还获得了在项目开始时没有预测的元胞自动机的重要结果。费米公式。费米子公式是由贝特安托组合数学导出的一个正整数系数多项式。证明了该多项式给出了仿射李代数的可积表示的分支函数，并利用量子群中的晶体理论给出了证明。我们也证明了这个猜想在几个情况下。Bethe Anastrous的组合数学在Bethe代数的组合研究中，除了费米子公式外，还有一个重要的代数方程系统，称为Q-系统。Kuniba与Nakanishi等人，在q = 0.3时，从Bethe方程得到了这个Q系统的解。孤子元胞自动机。虽然这方面的研究在我们最初并没有考虑到，但我们在孤子鞍自动机的研究中取得了新的进展。从量子仿射代数的有限维表示的晶体定义了元胞自动机。我们证明了在这种元胞自动机中孤子的运动可以分解为两体孤子的乘积，并且利用有限晶体的组合R可以明确地给出它们的散射规则.离散可积系统在仿射李代数An^<（1）>的情形下，上述孤立子元胞自动机是由非自治离散KP方程的超离散极限得到的。永井等人证明了孤子的性质，并从这种方法构造守恒量。