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.

谎言代数是描述某些物理系统对称性的数学结构。 许多这样的系统具有无限的独立对称性，因此自然研究是无限维的代数。 例如，一些最有趣的无限二维谎言代数，称为Aggine Kac-Moody Lie代数，描述了弦理论中的对称性。 与其他无限尺寸KAC-MOODY代数不同，仿射是从圆形到简单有限维谎言代数的功能空间的扩展。 这种几何解释是使Aggine在所有其他Kac-Moody代数中如此特别的代数的很大一部分。 它导致了与许多纯数学和粒子物理学领域的壮观相互作用，例如顶点操作员代数，可集成系统，量子群，结，结，模块化形式和完美的场理论。 在过去的几年中，数学家已经开始研究这种类型的其他代数，称为当前代数，基于来自更通用空间的函数代数，称为仿射方案，以有限的维度代数代数。 对于这些代数的知之甚少，尽管它们在物理和纯粹的数学中都非常自然，因为它们充当了几何空间的对称性，称为表示形式。 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亚词法的这些表示形式进行分类，这些代数由转换组留下。 与其他专家一起，我还将探索一种复杂的关系，称为古典世界和量子世界之间的Kazhdan-Lusztig等价，以及从其代表理论中重建当前代数。 我的学生将与相关的对称代数合作，称为Toroidal代数，超符号代数和Aggine W-Algebras。 这些中的每一个都是根据当前代数来定义的：圆环代数来自圆环的功能（数学甜甜圈），超符号代数通过数据附加到当前的保形超级级别的代数，而仿射W-algebras通过称为量子量化的量子量子构建了affine w-algebras。 该研究计划应更好地了解当前代数是什么，以及它们如何作为对称性。 它还应该通过受量子组启发的新技术来增强我们对仿射谎言代数的了解。 学生项目将与顶点操作员代数建立重要的联系，该链接可以看作是保形场理论的代数类似物。