Infinite dimensional Lie theory and representation theory

一百多年前，挪威数学家Sophus Lie引入了李论的基本思想，作为研究微分方程领域问题的一种手段。从那时起，李论已经成为当代数学的一个组成部分，它与其他几个领域相互作用，如微分几何、代数几何、谐波分析、数学物理、动力系统和数论，仅举几例。李论的基本对象是连续变换群（称为李群）和与李群相关的某些代数结构（称为李代数）。在物理和数学的许多分支中，李群和李代数可以用来描述和更好地理解连续对称性。一般来说，将李群与数学或物理系统的对称性联系起来的是一个代数结构，称为李群的表示（或与之相关的李代数）。例如，在现代物理学中，洛伦兹群的表示被用来理解基本粒子的各种性质。李理论和物理学之间现存的相互作用揭示了这个问题的另一个方面：无限维的李群和李代数与有限维的李群和李代数一样重要。在20世纪70年代，由于粒子物理学中超对称理论的出现，数学家和物理学家都被吸引去研究李群和李代数的梯度类似物的结构和表示，这些类似物现在被称为李超群和李超代数。这些分级概念的基础思想是允许可交换坐标和反可交换坐标。李超群和李超代数的引入导致了一个活跃的数学研究的全新领域。本文的目的是研究李代数/李群和李超代数/李超群的结构理论和表示理论。

More than a hundred years ago, the Norwegian mathematician Sophus Lie introduced the basic ideas of Lie theory as a means to study problems in the area of differential equations. From then on, Lie theory has emerged to an integral part of contemporary mathematics which interacts with several other areas, such as differential geometry, algebraic geometry, harmonic analysis, mathematical physics, dynamical systems, and number theory, to name a few. The fundamental objects of Lie theory are continuous transformation groups which are called Lie groups, and certain algebraic structures associated to Lie groups which are called Lie algebras. In many branches of physics and mathematics, Lie groups and Lie algebras can be used to describe and better understand continuous symmetries. Generally speaking, what links a Lie group to symmetries of a mathematical or a physical system is an algebraic structure which is called the representation of the Lie group (or its associated Lie algebra). For example, in modern physics, representations of the Lorentz group have been used to understand various properties of elementary particles. The existing interaction between Lie theory and physics reveals yet another aspect of the subject: that the infinite dimensional Lie groups and Lie algebras are as important as the finite dimensional ones. Motivated by the advent of supersymmetry as a theory in particle physics, in the 1970's mathematicians as well as physicists were drawn to study the structure and representations of graded analogues of Lie groups and Lie algebras which are nowadays called Lie supergroups and Lie superalgebras. The idea that underlies these graded concepts is to allow both commuting and anti-commuting coordinates. The introduction of Lie supergroups and Lie superalgebras has led to a whole new domain of active mathematical research. The goal of this proposal is to study the structure theory and representation theory of Lie algebras/Lie groups and Lie superalgebras/Lie supergroups.