What is a classical r-matrix?
What is a classical r-matrix?
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DOI：
10.1007/bf01076717
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发表时间：
1983
影响因子：
0.4
通讯作者：
M. A. Semenov-Tyan-Shanskii
中科院分区：
文献类型：
--
作者：
M. A. Semenov-Tyan-Shanskii
经典r矩阵方法作为量子逆散射方法的“副产品”出现在Sklyanin [1]和[2]的论文中，并且已经获得了相当大的普及。与经典矩阵相关的经典Yang~Baxter方程在[3-5]中有详细讨论。在本文中，我们建立了 r 矩阵方法和黎曼问题方法之间的联系，黎曼问题方法是目前可用于积分非线性方程的最有效工具。这使我们对杨-巴克斯特方程有了一种新的方法。我们的研究是仔细研究论文的结果[3-5]。 N. Yu的结果。 Reshetikhin 和 LD Eaddeev* 在阐明 r 矩阵的代数含义方面发挥了重要作用。 EK Sklyanin 和 AG Reiman 的建议对我准备本文很有帮助。对他们，我表示深深的谢意。我们处理各种主题的方法主要适合熟悉群上的哈密顿力学以及处理 Lax 形式可积方程的标准方法的数学家。一些初步结果发表在作者的笔记[7]中。简单介绍一下论文的结构。第 I-3 节是介绍性的。它们主要涉及将 r 矩阵方法中的标准张量符号转换为更通用的李代数数学语言。此外，还详细讨论了与论文[5]定义的联系。我们在第二节中做了一个有点矛盾的观察。 3 的缺点是，作为 r 矩阵方法基础的 Yang--Baxter 方程实际上从未在严格意义上得到满足（至少在存在适定黎曼问题的情况下）。修改后的 Yang--Baxter 方程，在第 2 节中介绍。 3、允许构建黎曼问题的抽象版本
The method of the classical r-matrix appeared in the papers of Sklyanin [l] and [2] as a" by-product" of the quantum inverse-scattering method and has already acquired considerable popularity. The classical Yang~ Baxter equation, which is connected with the classical rmatrices, was discussed in detail in [3-5]. In this paper we establish a connection between the r-matrix method and the Riemann-problem method, which is the most effective tool presently available for integrating nonlinear equations. This leads us to a new approach to the Yang--Baxter equation. Our study is the outcome of a careful study of the papers [3-5]. The results of N. Yu. Reshetikhin and LD Eaddeev* played an important role in clarifying the algebraic meaning of the r-matrix. The advice of EK Sklyanin and AG Reiman was helpful to me in the preparation of this paper. To them, I express my deep gratitude. Our approach to the various topics is mainly suitable for the mathematician who is familiar with Hamiltonian mechanics on groups and with the standard methods of handling integrable equations of Lax form. A number of preliminary results were published in the author's note [7].A few words on the structure of the paper. Sections I-3 are introductory. They deal mainly with the translation of the tensor notations, which are standard in the r-matrix method, into the more general mathematical language of the Lie algebras. Also, the connections with the definitions of paper [5] are discussed in detail. A somewhat paradoxical observation we make in Sec. 3 is that the Yang--Baxter equation, which lies at the basis of the r-matrix method, is never actually fulfilled in the rigorous sense (at least in those cases where a well-posed Riemann problem exists). The modified Yang--Baxter equation, introduced in Sec. 3, permits the construction of an abstract version of the Riemann-problem