Hyperbolic Kac-Moody Group Symmetry and Applications

这个项目是关于双曲型Kac-Moody群的，双曲型Kac-Moody群是已经被广泛研究的无限维李群。我们对含有仿射Kac-Moody子代数的无限维双曲Kac-Moody代数的李群很感兴趣。提出的一些问题是出于身体上的动机。我们对超重力等物理理论的研究揭示了一些引人入胜和耐人寻味的数学问题，这与卡克-穆迪群理论中的公开问题是一致的。这项拟议的工作是第一个旨在将双曲Kac-Moody群的对称性应用于理论物理中的超引力模型研究的数学创举。虽然双曲Kac-Moody对称性与超引力理论的一些对应是猜想的，但发展双曲Kac-Moody群及其可计算的对称空间的完整数学理论将对理解关于超引力对称群的公开问题有重要影响。本项目的目的是增进对高能论物理中代数对称性研究的理解。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。本课题研究包含仿射Kac-Moody子代数的无限维双曲Kac-Moody代数的李群。

This project concerns hyperbolic Kac-Moody groups, which are infinite dimensional Lie groups that have yet been extensively studied. We are interested in the Lie groups of infinite dimensional hyperbolic Kac-Moody algebras which contain affine Kac-Moody subalgebras. Some of the proposed problems are physically motivated. Our study of physical theories such as supergravity,a theory that incorporates both supersymmetry and general relativity, has revealed a number of compelling and intriguing mathematical problems, consistent with open problems in Kac-Moody group theory. The proposed work is the first mathematical initiative that aims to apply the symmetry properties of hyperbolic Kac-Moody groups to the study of supergravity models in theoretical physics. While some of these correspondences of hyperbolic Kac-Moody symmetry with supergravity theories are conjectural, developing a full mathematical theory of hyperbolic Kac-Moody groups and their symmetric spaces amenable to computation will have a significant impact on the understanding of open problems concerning the symmetry groups of supergravity.The objective of this project is to advance understanding in the study of algebraic symmetries underlying high energy theoretic physics. 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. This project concerns Lie groups of infinite dimensional hyperbolic Kac-Moody algebras which contain affine Kac-Moody subalgebras.