Lattices, Trees and Group Actions

本课程的目标是增进对代数、群论、表示论和组合学的理解，特别强调物理学和几何学的数学基础。几乎所有的有限维半单李群和李代数都发生在时空对称性和粒子物理标准模型的发展中，如果没有对对称性和群变换的理解，这是不可能取得进展的。无限维广义，称为Kac-Moody代数及其相关群，自然形成两个不同的类，即仿射和双曲。这类仿射Kac-Moody代数在基本粒子论、量子场论、规范理论、保形场理论、引力和弦理论等物理理论中有着广泛的应用。最近，双曲Kac-Moody群和代数的许多性质在高能物理中被发现，尽管这些物体的完整数学结构和物理环境的某些方面还没有被很好地理解。此外，与物理理论对应的数学证明是缺乏或不完整的，关于双曲代数和群的结构的基本问题仍然悬而未决。这在一定程度上是我们当前和拟议工作的动力。关于双曲Kac-Moody群和代数的结构的基本问题仍然没有得到回答。该提案使用所有可用的数学技术直接解决了这些问题。我们的工作还涉及到一些关于双曲Kac-Moody群和代数的数学问题，这些问题的动机是M-理论中Kac-Moody对称性的发现，超引力和降维，多维重力和宇宙台球的解。提出的研究符合以下长期目标：将Kac-Moody群、离散和连续建筑物的自同构群、Lie理论、表示理论和自同构形式进行比较，找到双曲Kac-Moody群的经典解释，使用Lorentz空间的几何以及C*-代数和非对易几何的方法来研究Kac-Moody群，为研究物理学中Kac-Moody对称的最新发现建立一个数学框架。我们最近的工作和建议结合了一系列数学技术，包括群论、代数、表示论、分析、几何、算术和组合方法，以及这些学科之间的相互作用。本课程的目标是增进对代数、群论、表示论和组合学的理解，特别强调物理学和几何学的数学基础。我们正在研究有限维对称的无限维推广，称为李群和李代数，它们出现在时空对称和粒子物理标准模型的发展中，并在数学的不同领域中广泛发挥作用。无限维广义，称为Kac-Moody结构，自然形成两个不同的类别，即仿射和双曲。20世纪80年代S提出的仿射Kac-Moody对称性在基本粒子理论、量子场论、规范理论、共形场理论、引力和弦理论等物理理论中有着广泛的应用。最近，双曲Kac-Moody对称的许多性质在高能物理中被发现，尽管这些天体的完整数学结构和物理环境的某些方面还没有被很好地理解。此外，与物理理论对应的数学证明是缺乏或不完整的，关于双曲Kac-Moody对称结构的基本问题仍然悬而未决。该提案使用所有可用的数学技术直接解决了这些问题。我们的工作还涉及一些关于双曲Kac-Moody结构的数学问题，这些问题的动机是M-理论中Kac-Moody对称性的发现(提出的5个弦理论的统一)、超引力和降维、多维重力解和宇宙台球。我们的目标是加强研究这些物理应用的数学框架，并确定和探索代数和几何中与当代高能物理发展相关的问题。

The objective of this program is to advance understanding in algebra, group theory, representation theory and combinatorics, with particular emphasis on the mathematics underlying physics and geometry. Almost all finite dimensional semisimple Lie groups and Lie algebras occur in space-time symmetries and the development of the Standard Model of particle physics, which could not have progressed without an understanding of symmetries and group transformations. Infinite dimensional generalizations, known as Kac-Moody algebras and their associated groups, naturally form two distinct classes, namely affine and hyperbolic. By the 1980's the class of affine Kac-Moody algebras was shown to have wide applications in physical theories such as elementary particle theory, quantum field theory, gauge theory, conformal field theory, gravity and string theory. Recently many properties of hyperbolic Kac-Moody groups and algebras have been discovered in high-energy physics, though the full mathematical structure of these objects and certain aspects of the physical settings are not well understood. Moreover, mathematical proofs of the correspondences with physical theories are lacking or incomplete, and fundamental questions about the structure of hyperbolic algebras and groups remain open. This in part serves as motivation for our current and proposed work. Fundamental questions about the structure of hyperbolic Kac-Moody groups and algebras remain unanswered. The proposal addresses these questions directly using all available mathematical techniques. Our work also involves a number of mathematical questions about hyperbolic Kac-Moody groups and algebras that are motivated by the discovery of Kac-Moody symmetry in M-theory, supergravity and dimensional reduction, solutions of multidimensional gravity and cosmological billiards. The research proposed is in line with the following long term goals: to draw parallels between Kac-Moody groups, automorphism groups of discrete and continuous buildings, Lie theory, representation theory and automorphic forms, to find a classical interpretation of hyperbolic Kac-Moody groups, to use the geometry of Lorentz space and the methods of C*-algebras and noncommutative geometry to study Kac-Moody groups, to build a mathematical framework for studying recent discoveries of Kac-Moody symmetries in physics. Our recent work and proposal incorporates a range of mathematical techniques including group theory, algebra, representation theory, analysis, geometry, arithmetic and combinatorial methods, as well as the interactions between these subjects. The objective of this program is to advance understanding in algebra, group theory, representation theory and combinatorics, with particular emphasis on the mathematics underlying physics and geometry. We are studying infinite dimensional generalizations of the finite dimensional symmetries, known as Lie groups and Lie algebras, which occur in space-time symmetries and the development of the Standard Model of particle physics, and are widespread in their roles in diverse areas of mathematics. Infinite dimensional generalizations, known as Kac-Moody structures, naturally form two distinct classes, namely affine and hyperbolic. By the 1980's affine Kac-Moody symmetries were shown to have wide applications in physical theories such as elementary particle theory, quantum field theory, gauge theory, conformal field theory, gravity and string theory. Recently many properties of hyperbolic Kac-Moody symmetries have been discovered in high-energy physics, though the full mathematical structure of these objects and certain aspects of the physical settings are not well understood. Moreover, mathematical proofs of the correspondences with physical theories are lacking or incomplete, and fundamental questions about the structure of hyperbolic Kac-Moody symmetry structures remain open. The proposal addresses these questions directly using all available mathematical techniques. Our work also involves a number of mathematical questions about hyperbolic Kac-Moody structures that are motivated by the discovery of Kac-Moody symmetry in M-theory, (the proposed unification of 5 string theories), supergravity and dimensional reduction, solutions of multidimensional gravity and cosmological billiards. Our objective is to strengthen the mathematical framework for studying these physical applications, and to identify and explore problems in algebra and geometry that are of relevance to the development of high-energy contemporary physics.