Elliptic quantum groups, deformed W algebras of type D and their applications to the analysis of Baxter's eight-vertex model

椭圆量子群、D型变形W代数及其在巴克斯特八顶点模型分析中的应用

• 批准号: 16540183
• 负责人: SHIRAISHI Jun'ichi
• 金额: $ 2.48万
• 依托单位: The University of Tokyo
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2004
• 资助国家: 日本
• 起止时间: 2004 至 2007
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-16540183/
• 关键词: deformed W-algebra; elliptic quantum group; Macdonald polynomial; Koormwinder polynomial; Baxter's eight-vertex model; vertex operator; hypergeometric series; Hirota-Miwa equation; Macdonald多項式; Koornwinder多項式; 多変数超幾何級数; 可解格子模型; 相関関数; 8頂点模型 /

The aim of this project is to study the correlation functions for Baxter's eight-vertex model, by applying the representation theory of the deformed W-algebras of type D. Here, the deformed W-algebras are certain variation of the so-called elliptic quantum groups, which are defined by specifying a set of elliptic functions (structure functions). I summarize my results in what follows.(1) Using the deformed W-algebras, the vertex operators for Baxter's eight-vertex model are explicitly constructed. The rank of the W-algebra is determined by the arithmetic property of the so-called crossing parameter of the model.(2) It is conjectured that the matrix elements of the vertex operators are uniquely characterized by a certain integral transformation, which commutes with the action of the Macdonald difference operators. We can regard the problem a discrete analogue of the initial value problem associated with integrable dynamical systems.(3) To find a proof of the above conjecture, I studied … More the eigenvalue problem associated with the integral operator, and the Macdonald difference operators acting on the space of formal power series (not on the space of the symmetric polynomials). Not only the case of A-type root lattice, I treated the case of B, C, D, and the BC case, too. I found several quite nontrivial explicit formulas for such series by using computer algebra. Hence, I started to have communication with M. Noumi and found the kernel functions for the Koornwinder difference operator. As an application we found explicit formulas for the Koornwinder polynomials associated with single row and single column cases(4) I studied the integrals of motion associated with the deformed W-algebras. By taking a classical limit, I derived an integrable hierarchy whish is described by the Sato theory with a certain reduction condition. The tau-function of this system satisfied a version of the Hirota-Miwa bilinear equation. In some degeneration limits, I found that a class of special solutions to the bilinear equation can be constructed, and the corresponding integrals of motion can be explicitly calculated. The results indicate that the classical mechanical object have a good amount of information which Macdonald polynomials or Hall-Littlewood polynomials have. Less

该项目的目的是通过应用 D 型变形 W 代数的表示论来研究 Baxter 八顶点模型的相关函数。这里，变形 W 代数是所谓的椭圆量子的某些变体群，它们是通过指定一组椭圆函数（结构函数）来定义的。我在下面总结了我的结果。（1）使用变形的 W 代数，顶点算子Baxter的八顶点模型被明确地构造出来，W-代数的秩由模型的所谓交叉参数的算术性质决定。(2)推测顶点算子的矩阵元素具有唯一的特征。通过一定的积分变换，它与麦克唐纳差分算子的作用进行交换。我们可以将这个问题视为与可积动力系统相关的初始值问题的离散模拟。（3）找到上述问题的证明。猜想，我研究了与积分算子相关的特征值问题，以及作用于形式幂级数空间（而不是对称多项式空间）的麦克唐纳差分算子，不仅是 A 型根格的情况，我也处理了 B、C、D 的情况，以及 BC 的情况，我通过计算机代数找到了几个非常重要的显式公式，因此，我开始与 M. Noumi 进行交流，并找到了作为一个应用，我们发现了与单行和单列情况相关的 Koornwinder 多项式的显式公式 (4) 我研究了与变形 W 代数相关的运动积分。我推导了佐藤理论在一定的简化条件下描述的可积层次结构，该系统的 tau 函数满足 Hirota-Miwa 双线性方程的某种退化。极限，我发现可以构造一类双线性方程的特殊解，并且可以显式计算相应的运动积分。结果表明，经典机械对象具有麦克唐纳多项式或 Hall-Littlewood 的大量信息。多项式有。

项目成果

Koornwinderのq差分作用素の核函数とその応用

Koornwinder的q差分算子核函数及其应用

• 发表时间: 2007
• 作者: M.Saigo;Sh.Owa;V.Kiryakova;野海 正俊
• 通讯作者: 野海 正俊

Kernel function for Koornwinder's operator and their applications

Koornwinder 算子的内核函数及其应用

• 发表时间: 2007
• 作者: NOUMI;Masatoshi;SHIRAISHI;Jun'ichi
• 通讯作者: Jun'ichi

Free field constructions for the elliptic algebra ${\calA}_{q, p}(\widehat{ sl}_2)$ and Baxter's eight-vertex model

椭圆代数 ${calA}_{q, p}(widehat{ sl}_2)$ 和 Baxter 八顶点模型的自由场构造

• 发表时间: 2004
• 期刊: Proceedings of 6th International Workshop on Conformal Field Theory and Integrable Models. Intemat. J. Modern Phys. A 19 May, suppl.
• 作者: SHIRAISHI;Jun'ichi
• 通讯作者: Jun'ichi

Deformed W-algebra and correlation functions for Baxter's eight vertex model

Baxter 八顶点模型的变形 W 代数和相关函数

• 发表时间: 2005
• 作者: Sh.Owa;M.Acu;白石潤
• 通讯作者: 白石潤

A Conjecture about Raising Operators for Macdonald Polynomials

麦克唐纳多项式算子的提升猜想

• 发表时间: 2005
• 期刊: Letters in Mathematical Physics Volume 73, Number 1
• 作者: Arnaudon;Daniel;Avan;Jean;Frappat;Luc;Ragoucy;Eric;Shiraishi;Junichi;Jun'ichi Shiraishi et al.;Saburo Kakei;Jun'ichi Shiraishi
• 通讯作者: Jun'ichi Shiraishi