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

本课题的目的是应用d型变形w -代数的表示理论，研究Baxter八顶点模型的相关函数。这里，变形w -代数是所谓椭圆量子群的某种变异，它是通过指定一组椭圆函数（结构函数）来定义的。我将我的结果总结如下。(1)利用变形w代数，明确构造了Baxter八顶点模型的顶点算子。w代数的秩是由模型的所谓交叉参数的算术性质决定的。(2)推测顶点算子的矩阵元素具有一定的积分变换的唯一特征，该变换与麦克唐纳差分算子的作用交换。我们可以把这个问题看作与可积动力系统有关的初值问题的离散模拟。(3)为了找到上述猜想的证明，我进一步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Macdonald差分算子。不仅是a型根格的情况，我还处理了B、C、D和BC的情况。我用计算机代数找到了几个非常重要的显式公式。于是，我开始和M. Noumi交流，找到了Koornwinder差分算子的核函数。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。在经典极限条件下，导出了用佐藤理论描述的具有一定约化条件的可积层次。该系统的τ函数满足Hirota-Miwa双线性方程的一个版本。在某些退化极限下，我发现双线性方程的一类特解可以构造出来，相应的运动积分可以显式计算出来。结果表明，经典力学对象具有Macdonald多项式或Hall-Littlewood多项式所具有的大量信息。少

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