Extended affine Lie algebras, groups and representation theory

扩展仿射李代数、群和表示论

• 批准号: 8836-2011
• 负责人: Neher, Erhard
• 金额: $ 1.09万
• 依托单位: University of Ottawa
• 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=568619
• 关键词: Extended; affine; Lie; algebras; groups

Since the 19th century the mathematical concept of a group has been one of the most basic abstract structures used by mathematicians to describe symmetry arising in many different incarnations, e.g. in science and engineering. Mathematicians have developed a sophisticated theory for groups with more and more profound applications. The applications of groups within and outside of mathematics have been a remarkable success story. The most important type of groups originates from the fundamental work of the Norwegian mathematician Sophus Lie (1842--1899). These so-called Lie groups and their algebraic analogues, the algebraic groups, are distinguished by the fact that one can associate another mathematical structure, a Lie algebra, to them. Roughly speaking, Lie algebras are first order approximations of the corresponding groups. My research is concerned with certain types of Lie algebras and some related structures. In the first part of the 20th century, Cartan, Weyl, Jacobson and Chevalley have created a magnificent theory of finite-dimensional semisimple Lie algebras. Their theory was later extended to certain infinite-dimensional Lie algebras by Kac and Moody. In particular, the so-called affine Kac-Moody algebras have been a central topic of research in Lie algebras with many profound applications. Motivated by applications in quantum gauge theory and singularity theory, affine Kac-Moody algebras have been generalized to so-called extended affine Lie algebras. This new class of Lie algebras offers exciting new possibilities. It provides us with many more structures and hence possible applications than what was previously known. The goal of my research is to develop the structure and representation theory of extended affine Lie algebras and some other related Lie algebras. I will also study the groups associated to these Lie algebras and another algebraic structure, so-called Jordan algebras.

自19世纪以来，群的数学概念一直是数学家用来描述对称性的最基本的抽象结构之一。数学家们已经发展出了一种复杂的群理论，其应用越来越广泛。数学内外的群体应用是一个了不起的成功故事。最重要的一类群起源于挪威数学家Sophus Lie（1842- 1899）的基本工作。这些所谓的李群和它们的代数类似物，代数群的区别在于，人们可以将另一种数学结构，李代数，与它们联系起来。粗略地说，李代数是相应群的一阶近似。我的研究涉及某些类型的李代数和一些相关的结构。 世纪上半叶，Cartan、Weyl、Jacobson和Chevalley等人建立了一个宏伟的有限维半单李代数理论。他们的理论后来被Kac和Moody扩展到某些无限维李代数。特别是，所谓的仿射Kac-Moody代数一直是李代数研究的中心课题，具有许多深刻的应用。 受量子规范理论和奇点理论的启发，仿射Kac-Moody代数被推广到所谓的扩展仿射李代数。这一类新的李代数提供了令人兴奋的新可能性。它为我们提供了更多的结构，因此可能的应用比以前已知的。本论文的主要研究目标是发展广义仿射李代数及相关李代数的结构和表示理论。我还将研究与这些李代数和另一个代数结构，所谓的约旦代数相关的群。