Representations in Infinite-Dimensional Lie Theory

李代数是描述某些物理系统的对称性的数学结构。许多这样的系统都有无限多个独立的对称，所以研究无限维李代数是很自然的。例如，一些最有趣的无限维李代数，称为仿射Kac-Moody李代数，描述了弦理论中的对称性。*与其他无限维Kac-Moody李代数不同，仿射李代数是函数空间从圆到单有限维李代数的扩张。这种几何解释是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原因。它已经与纯数学和粒子物理学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、纽结不变量、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其他这种类型的李代数，称为当前代数，基于来自更一般空间的函数的代数，称为仿射方案，到有限维李代数。对这些代数的了解要少得多，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性，称为表示。*确定哪种对称是最重要的之后，我计划对当前代数及其子代数的这些表示进行分类，这些表示通过变换组保持不变。我还将与其他专家一起探索经典世界和量子世界之间一种称为Kazhdan-Lusztig等价的复杂关系，以及从它们的表示理论重构当前代数。我的学生将学习相关的对称代数，称为环状代数、超共形代数和仿射W-代数。其中每一个都是用现在的代数来定义的：环状代数来自环面(一个数学甜甜圈)上的函数，超共形代数是由共形超代数的当前代数上的数据描述的，仿射W-代数是由仿射李代数通过一个称为量子哈密尔顿约化的过程构造的。*研究计划应该会导致更好地理解什么是当前代数，以及它们是如何作为对称的。它还应该通过受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符代数建立重要的联系，顶点算符代数可以被视为共形场理论的代数类似物。*****

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. *****