Multi-affine Lie algebras and their representations

多重仿射李代数及其表示

• 批准号: RGPIN-2015-05967
• 负责人: Gao, Yun
• 金额: $ 1.02万
• 依托单位: York University
• 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=590755
• 关键词: Multi; affine; Lie; algebras; their

Extended affine Lie algebras are a natural generalization of the affine Kac-Moody Lie algebras. They were introduced by mathematical physicists in understanding the high dimensional string theory and conformal field theory. This class of Lie algebras is closely related to the extended affine root systems of Saito, the generalized intersection matrix  Lie algebras of Slodowy, and the root graded Lie algebras studied by Berman-Moody, Benkart-Zelmanov, Neher, among others. Every extended affine Lie algebra is associated with a finite irreducible root system (possibly non-reduced) and a nonnegative integer called nullity. The nullity zero yields exactly finite dimensional simple Lie algebras while the nullity one yields exactly affine Kac-Moody Lie algebras. The nullity greater than one yields multi-affine Lie algebras which include many interesting examples such as toroidal Lie algebras, and certain Lie algebras coordinated with quantum tori and even non-associative tori. The structure theory of the extended affine Lie algebras has been well understood. However, the representation theory of the extended affine Lie algebras is hard and far from complete. In this proposal, we will continue to study the extended affine Lie algebras, their representations and quantizations. We intend to explore some related algebras such as multi-affine Lie super-algebras as well as certain GIM Lie algebras. The following three topics are our interests for a short or longer term.

扩展仿射李代数是仿射 Kac-Moody 李代数的自然推广，它们是数学物理学家在理解高维弦理论和共形场论时引入的，这类李代数与扩展仿射根系密切相关。 Saito，Slodowy 的广义交集矩阵李代数，以及 Berman-Moody 研究的根分级李代数， Benkart-Zelmanov、Neher 等人提出，每个扩展的仿射李代数都与一个有限的不可约根系统（可能是非约简的）和一个称为零值的非负整数相关联，零值产生精确的有限维简单李代数，而零值则产生有限维简单李代数。精确仿射 Kac-Moody 李代数 大于 1 的零度产生多重仿射李代数，其中包括。许多有趣的例子，例如环形李代数，以及某些与量子环面甚至非结合环面协调的李代数。 扩展仿射李代数的结构理论已被很好地理解。 然而，扩展仿射李代数的表示理论很困难并且远未完成。在本提案中，我们将继续研究扩展仿射李代数、它们的表示和量化，我们打算探索一些相关的代数，例如多重仿射。李超代数以及某些 GIM 李代数 以下三个主题是我们短期或长期的兴趣。