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.

谎言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具有无限的独立对称性，因此研究无限的尺寸谎言代数非常重要。 例如，一些最有趣的无限尺寸谎言代数，称为Aggine Kac-Moody代数，描述了弦理论中的对称性。 比谎言代数本身更重要的是它们充当几何空间的对称性（称为表示形式）的方式。 谎言代数通常具有很多表示，并且理解它们提供了有关环境几何和物理背景的信息。 我们对谎言代数的家庭特别感兴趣，其中包括，但也超越了Aggine Kac-Moody代数。 像这样的代数有助于我们更好地了解仿射Kac-Moody代数周围的几何形状，并给出更复杂的更高维度结构的提示。 我们的工作有望确定具有某些有用特性的所有谎言代数，以提供一个共同的框架来研究两个不同的代数代数家族的表示，并开发新技术以理解与粒子物理学相关的无限尺寸几何形状。