Representations in Infinite Dimensional Lie Theory

无限维李理论中的表示

• 批准号: 341752-2012
• 负责人: Lau, Michael
• 金额: $ 1.09万
• 依托单位: Université Laval
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=572355
• 关键词: Representations; Infinite; Dimensional; Lie; Theory

Lie algebras are mathematical structures that describe symmetries of certain physical systems. Since many such systems have infinitely many independent symmetries, it is important to study infinite dimensional Lie algebras. For example, some of the most interesting infinite dimensional Lie algebras, called affine Kac-Moody algebras, describe symmetries in string theory. Even more important than the Lie algebras themselves are the ways in which they act as symmetries of geometric spaces, called representations. A Lie algebra typically has a great many representations, and understanding them gives information about the ambient geometric and physical context. We are particularly interested in families of Lie algebras that include, but also go beyond, the affine Kac-Moody algebras. Algebras like these help us better understand the geometry around affine Kac-Moody algebras, as well as give hints of more complex higher dimensional structures. Our work is expected to identify all the Lie algebras with certain useful properties, to provide a common framework for studying the representations of two different families of Lie algebras, and to develop new techniques for understanding infinite dimensional geometry related to particle physics.

李代数是描述某些物理系统对称性的数学结构。 由于许多这样的系统有无穷多个独立的对称，所以研究无限维李代数是很重要的。 例如，一些最有趣的无穷维李代数，称为仿射卡茨-穆迪代数，描述了弦论中的对称性。 比李代数本身更重要的是它们作为几何空间的对称性的方式，称为表示。 李代数通常有很多表示，理解它们可以提供有关周围几何和物理环境的信息。 我们特别感兴趣的家庭李代数，包括，但也超越，仿射卡茨-穆迪代数。 像这样的代数帮助我们更好地理解仿射Kac-Moody代数周围的几何，以及给出更复杂的高维结构的暗示。 我们的工作预计将确定所有的李代数与某些有用的属性，提供一个共同的框架，研究两个不同的家庭的李代数的表示，并开发新的技术，了解与粒子物理学有关的无限维几何。