Representations in Infinite-Dimensional Lie Theory

无限维李理论中的表示

• 批准号: RGPIN-2017-04280
• 负责人: Lau, Michael
• 金额: $ 1.17万
• 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: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=710459
• 关键词: Representations; Infinite; Dimensional; Lie; Theory

Lie algebras are mathematical structures that describe symmetries of certain physical systems. Many such systems have infinitely many independent symmetries, so it is natural to study Lie algebras which are infinite-dimensional. For example, some of the most interesting infinite-dimensional Lie algebras, called affine Kac-Moody Lie algebras, describe symmetries in string theory. Unlike other infinite-dimensional Kac-Moody algebras, the affine ones are extensions of spaces of functions from the circle to simple finite-dimensional Lie algebras. This geometric interpretation is a large part of what makes affine Lie algebras so special among all other Kac-Moody algebras. It has led to spectacular interactions with many areas of pure mathematics and particle physics, such as vertex operator algebras, integrable systems, quantum groups, knot invariants, modular forms, and conformal field theory. In the last few years, mathematicians have begun studying other Lie algebras of this type, called current algebras, based on algebras of functions from more general spaces, called affine schemes, to finite-dimensional Lie algebras. Much less is known about these algebras, though they appear very naturally in both physics and pure mathematics because of the beautiful ways in which they act as symmetries of geometric spaces, called representations. After determining which kinds of symmetries are the most important, I plan to classify these representations of current algebras and their subalgebras left invariant by groups of transformations. Together with other experts, I will also explore a complex relationship called Kazhdan-Lusztig equivalence between the classical and quantum world, and the reconstruction of current algebras from their representation theory. My students will work with related symmetry algebras, called toroidal algebras, superconformal algebras, and affine W-algebras. Each of these is defined in terms of current algebras: toroidal algebras come from functions on a torus (a mathematical doughnut), superconformal algebras are described by data attached to current algebras of conformal superalgebras, and affine W-algebras are constructed from affine Lie algebras via a process called quantum hamiltonian reduction. The research program should lead to a better understanding of what current algebras are, and how they act as symmetries. It should also enhance our knowledge of affine Lie algebras through new techniques inspired by quantum groups. The student projects will make important links with vertex operator algebras, which can be viewed as algebraic analogues of conformal field theory.

李代数是描述某些物理系统对称性的数学结构。 许多这样的系统有无穷多个独立的对称，所以研究无限维的李代数是很自然的。 例如，一些最有趣的无限维李代数，称为仿射卡茨-穆迪李代数，描述了弦论中的对称性。 与其他无限维Kac-Moody代数不同，仿射代数是函数空间从圆到简单有限维李代数的扩展。 这种几何解释是使仿射李代数在所有其他Kac-Moody代数中如此特殊的很大一部分。 它与纯数学和粒子物理的许多领域产生了惊人的相互作用，如顶点算子代数、可积系统、量子群、结不变量、模形式和共形场论。 在过去的几年里，数学家们已经开始研究这种类型的其他李代数，称为当前代数，基于更一般空间的函数代数，称为仿射方案，到有限维李代数。 我们对这些代数知之甚少，尽管它们在物理学和纯数学中都很自然地出现，因为它们以美丽的方式充当几何空间的对称，称为表示。 在确定了哪种对称性是最重要的之后，我计划对当前代数及其子代数的这些表示进行分类，这些表示通过变换群保持不变。 与其他专家一起，我还将探索经典和量子世界之间的复杂关系，称为Kazhdan-Lusztig等价，以及从其表示理论重建当前代数。 我的学生将研究相关的对称代数，称为环形代数，超共形代数和仿射W-代数。 其中每一个都是用当前代数来定义的：环面代数来自环面上的函数（数学甜甜圈），超共形代数由附在共形超代数的当前代数上的数据来描述，仿射W-代数通过称为量子哈密顿约化的过程从仿射李代数构造。 该研究计划应该导致更好地理解当前代数是什么，以及它们如何作为对称。 它还应该通过量子群启发的新技术来增强我们对仿射李代数的知识。 学生的专题将与顶点算子代数建立重要的联系，顶点算子代数可以被看作是共形场论的代数类似物。