Infinite dimensional lie algebras and their coordinate algebras

无限维李代数及其坐标代数

• 批准号: 8465-2006
• 负责人: Allison, Bruce
• 金额: $ 0.72万
• 依托单位: University of Victoria
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2007
• 资助国家: 加拿大
• 起止时间: 2007-01-01 至 2008-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=362115
• 关键词: Infinite; dimensional; lie; algebras; their

Finite dimensional simple Lie algebras, introduced by Sophus Lie over a hundred years ago, are of fundamental importance in geometry and physics as a measure of symmetry.  About 30 years ago, Victor Kac and Robert Moody independently introduced some infinite dimensional analogs, now called affine Kac-Moody algebras, of finite dimensional simple Lie algebras.  A rich mathematical theory of affine Kac-Moody Lie algebras has been developed since that time.  Furthermore, these algebras have been used in physics to study the symmetries that occur in field theories that model interactions of elementary particles. In 1990 a new class of Lie algebras, called extended affine Lie algebras, was introduced by two mathematical physicists Høegh-Krohn and Torresani.  This new class includes all finite dimensional simple Lie algebras and all affine Kac-Moody Lie algebras.  Remarkable progress has been made in the last decade in the study of the structure of these algebras, but many interesting questions remain to be investigated.  In the proposed research, I will continue my study of the structure of extended affine Lie algebras along with some related classes of algebras. One of my approaches will be to investigate realizations of these Lie algebras. Realizations allow one to make explicit calculations involving the algebras. One type of realization is as matrices with coordinates from an algebra called a coordinate algebra.  Part of my investigation will be devoted to the study of these coordinate algebras.

有限维单李代数是由Sophus Lie在一百多年前提出的，在几何学和物理学中具有重要的对称性。大约30年前，Victor Kac和Robert Moody独立地引入了有限维单Lie代数的一些无限维类比，现在称为仿射Kac-Moody代数。从那时起，仿射Kac-Moody李代数的丰富的数学理论已经发展起来。此外，这些代数已经被用于研究发生在场论中的对称性，这些场理论模拟基本粒子的相互作用。1990年，两位数学物理学家Höegh-Krohn和Torresani引入了一类新的李代数，称为扩展仿射李代数。这类新的李代数包括所有有限维单李代数和所有仿射Kac-Moody李代数。在过去的十年里，对这些代数的结构的研究取得了显著的进展，但仍有许多有趣的问题有待研究。在拟议的研究中，我将继续研究扩展仿射李代数的结构以及一些相关的代数类。我的方法之一将是研究这些李代数的实现。实现允许一个人进行涉及代数的显式计算。一种类型的实现是用来自称为坐标代数的代数的坐标作为矩阵。我的研究的一部分将致力于对这些坐标代数的研究。