Study of integrable structure inquanteem integrable model

关于可积点阵模型的可积结构，我们得到了以下几个新的结果。xxz型自旋模型Bethe方程的形式完备性定理。（Nakanishi, Kuniba, Tsuboi (Tokyo Univ)）通过对q=0极限下xxz型自旋模型的Bethe方程的弦解进行分类，证明了在量子仿射代数的KR模的Kirillov-Reshetikhin （KR）猜想下，表示xxz型自旋模型的形式完备性的KR模特征的幂级数公式对任何仿射李代数都成立。用q -系统的正则解重新表述Kirillov-Reshetikhin猜想。（Nakanishi, Kuniba, Tsuboi (Tokyo Univ.)）通过观察在研究1中得到的各种公式和定理的原理机制，我们完全阐明了这些表示形式完备性的幂级数公式与底层泛函（代数）方程（kr型q系）之间的关系。即，我们引入一类有限变量有限函数的泛函方程（有限q系统）。有限q系统有一个显著的性质，即它的唯一解允许用多个变量的拉格朗日反演公式得到两个幂级数公式。它们完全是xxx型和xxz型模型的形式完备性公式的有限变量模拟。形式完备性公式可作为该公式的投影极限得到。

We obtain the following new results on the integrable structure of integrable lattice models.1. The formal completeness theorem on the Bethe equation for the XXZ-type spin models. (Nakanishi, Kuniba, Tsuboi (Tokyo Univ.))By classifying the string solutions of the Bethe equation for the XXZ-type spin models in the q=0 limit, we showed that, under the Kirillov-Reshetikhin (KR) conjecture on the KR modules of the quantum affine algebras, the power series formula of the character of the KR module representing the formal completeness of the XXZ-type spin models holds for any affine Lie algebra.2. Reformulation of the Kirillov-Reshetikhin conjecture by the canonical solutions of the Q-systems. (Nakanishi, Kuniba, Tsuboi (Tokyo Univ.))By observing the principle mechanism for various formulae and theorems obtained in the study of 1, we completely clarified the relation between these power series formulae representing the formal completeness and the underlying functional (algebraic) equations (Q-system of KR-type). Namely, we introduce a kind of functional equations (finite Q-system) for a finite number of functions of a finite number of variables. A finite Q-system has a remarkable property that its unique solution admits two power series formula by the Lagrange inversion formula for several variables. They are exactly a finite-variable analogue of the formal completeness formulae for the XXX-type and XXZ-type models. The formal completeness formula can be obtained as the projective limit of this formula.