Hyperbolic Kac-Moody Group Symmetry and Applications

该项目涉及双曲Kac-Moody群，这是一种尚未被广泛研究的无限维李群。研究了包含仿射Kac-Moody子代数的无限维双曲Kac-Moody代数的李群。提出的一些问题是由生理因素引起的。我们对物理理论的研究，如超引力，一个结合了超对称和广义相对论的理论，揭示了许多引人注目和有趣的数学问题，与Kac-Moody群论中的开放问题一致。提出的工作是第一个数学倡议，旨在将双曲Kac-Moody群的对称性应用于理论物理中超重力模型的研究。虽然一些双曲Kac-Moody对称与超重力理论的对应关系是推测性的，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有关超重力对称群的开放问题产生重大影响。本计画的目标是促进对高能理论物理基础上的代数对称性研究的理解。几乎所有有限维半单李群和李代数都发生在时空对称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，这一发展就不可能取得进展。无限维推广，被称为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。到20世纪80年代，仿射Kac-Moody代数被证明在基本粒子理论、量子场论、规范理论、共形场论、重力和弦理论等物理理论中具有广泛的应用。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.