Affine Lie algebra characters and Bethe Ansatz

在本研究中，我们使用可解晶格模型中的Bethe Ansatz方法研究了仿射李代数特征。我们还在元胞自动机方面取得了重要的成果，这是项目开始时没有预测到的。费密子公式。费米子公式是由贝特·安萨茨的组合学产生的一个系数为正整数的多项式。我们推测这个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考虑了它的证据。我们还用几个例子证明了这个猜想。Bethe Ansatz的组合。在贝特·安萨茨的组合研究中，除了费米子公式之外，还有一个重要的代数方程组，称为q系统。Kuniba和Nakanishi等人从Bethe方程中得到了q = 0.3时该q -system的解。孤子元胞自动机。虽然我们一开始并没有想到这项研究，但我们在孤子自动机的研究上有了新的进展。元胞自动机是由量子仿射代数的有限维表示晶体定义的。我们证明了该元胞自动机中孤子的运动被分解为二体孤子的积，并利用有限晶体的组合R明确给出了它们的散射规律。离散可积系统。上面提到的仿射李代数的孤子元胞自动机An^<(1)>是由非自治离散KP方程的超离散极限得到的。Nagai等人用这种方法证明了孤子性质并构造了守恒量。

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.