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 /; 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