Study of integrable structure inquanteem integrable model

Quanteem可积模型中的可积结构研究

• 批准号: 10640016
• 负责人: NAKANISHI Tomoki
• 金额: $ 2.05万
• 依托单位: Nagoya University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1998
• 资助国家: 日本
• 起止时间: 1998 至 2001
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640016/
• 关键词: Bethe ansatz; Heisenberg model; quantum group; intergrable models; ベーテ方程式; ベーテ仮説; Bethe方程式; 可積分格子模型

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.

我们得到了以下关于可积格模型的可积结构的新结果. XXZ型自旋模型Bethe方程的形式完备性定理。（中西、国场、坪井（东京大学））通过对Bethe方程在q=0极限下XXZ型自旋模型的弦解进行分类，证明了在量子仿射代数KR模的Kirillov-Reshetikhin（KR）猜想下，表示XXZ型自旋模型形式完备性的KR模特征标的幂级数公式对任意仿射李代数成立.用Q-系统的正则解重新表述Kirillov-Reshetikhin猜想。（中西、国场、坪井（东京大学））通过观察文献[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其基础的泛函（代数）方程（KR型Q-系统）之间的关系。也就是说，我们引进了一类函数方程（有限Q-系统）的有限个函数的有限个变量。有限Q-系统有一个显著的性质，即它的解存在两个幂级数公式，这是由多元的拉格朗日反演公式得到的。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。形式完备性公式可以作为该公式的投影极限得到。

项目成果

T.Nakanishi, (A.Kuniba, Z.Tsuboi): "The Bethe equation at q=0, the Mobius inversion formula, and areight Multiplicities : III The Xn^r case"Letters in Mathematical Physics. 印刷中.

T. Nakanishi，（A.Kuniba，Z.Tsuboi）：“q=0 时的贝特方程、莫比乌斯反演公式和正重数：III Xn^r 情况”数学物理学快报正在出版。

T.Nakanishi(with A.Kuniba, Z.Ysuboi): "The Bethe equation at q=0, the Mobius inversion formula, and weight multiplicities : III THE X_n case"Letters in Mathematical Physics. (in press).

T.Nakanishi（与 A.Kuniba、Z.Ysuboi）：“q=0 时的贝特方程、莫比乌斯反演公式和权重重数：III X_n 情况”数学物理学快报。

T.Nakanishi, A.Kuniba, Z.Tsuboi: "The Bethe equation at q=0, the Mobius inversion formula, and weight multiplicities : III The X^<(r)>_n case"Letters in Mathematical Physics. (印刷中).

T.Nakanishi、A.Kuniba、Z.Tsuboi：“q=0 时的贝特方程、莫比乌斯反演公式和权重重数：III X^<(r)>_n 情况”《数学物理学快报》（正在出版）。 ）。

T.Nakanishi, (A.Kuniba, Z.Tsuboi): "The canonical sdutions of the Q-systems and the Kirillov-Reshetikhin conjecture"Communications in Mathematical Physics. 印刷中.

T. Nakanishi，（A.Kuniba，Z.Tsuboi）：“Q 系统和基里洛夫-列谢蒂欣猜想的规范研究”数学物理通讯出版。

A.Kuniba,T.Nakanishi: "The Bethe equation at 9=0,the Mobius inversion formula, and weight multiplicities II, Xn case "J.Algebra. (掲載予定).

A.Kuniba、T.Nakanishi：“9=0​​ 时的 Bethe 方程、莫比乌斯反演公式和权重多重性 II，Xn 情况”J.Algebra。