Integrable Systems and Factorization Problems
Integrable Systems and Factorization Problems
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DOI：
10.1007/978-3-0348-8003-9_4
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发表时间：
2002-09
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影响因子：
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通讯作者：
M. Semenov-Tian-Shansky
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作者：
M. Semenov-Tian-Shansky
本讲座是为2000年9月的”Faro国际分解和可积系统暑期学校”编写的。他们的目的是为参与者的背景分析和算子理论，但没有特殊知识的几何和李群。下面的文字代表了一种妥协：完全不谈论李代数和李群当然是不可能的;然而，为了使主要思想合理地清楚，我试图只使用矩阵代数，如g [（n）]和它的自然子代数;所用的李群是GL（n）及其子群，或者由圆上的矩阵值函数组成的循环群（可能允许黎曼球面的部分扩展）。我希望这能让分析师的生活环境变得足够容易。主要目标是解释如何因式分解问题（通常是矩阵黎曼问题）产生整个小世界的可积系统沿着几何的相空间，哈密顿结构，拉克斯表示，运动积分和明确的解决方案。关键的工具将是经典的r-矩阵（一个对象，其另一个伪装是众所周知的希尔伯特变换）。我不会给出技术细节，除非它们可以在几行中暴露出来;另一方面，只要有可能，所有的动机都是完整的。我希望这一选择同意的精神，法鲁学校，并将有助于弥合差距之间的不同分支的数学分析。
The present lectures were prepared for the" Faro International Summer School on Factorization and Integrable Systems" in September 2000. They were intended for participants with the background in analysis and operator theory but without special knowledge of geometry and Lie groups. The text below represents a sort of compromise: it is certainly impossible not to speak about Lie algebras and Lie groups at all; however, in order to make the main ideas reasonably clear, I tried to use only matrix algebras such as g [(n) and its natural subalgebras; Lie groups used are either GL (n) and its subgroups, or loop groups consisting of matrix-valued functions on the circle (possibly admitting an extension to parts of the Riemann sphere). I hope this makes the environment sufficiently easy to live in for an analyst. The main goal is to explain how the factorization problems (typically, the matrix Riemann problem) generate the entire small world of integrable systems along with the geometry of the phase space, Hamiltonian structure, Lax representations, integrals of motion and explicit solutions. The key tool will be the classical r-matrix (an object whose other guise is the well-known Hilbert transform). I do not give technical details, unless they may be exposed in a few lines; on the other hand, all motivations are given in full scale whenever possible. I hope that this choice agrees with the spirit of the Faro School and will help to bridge the gap between different branches of Mathematical analysis.