Representations in infinite-dimensional lie theory

李代数是描述某些物理系统对称性的数学对象。由于许多这样的系统具有无限多个独立的对称性，因此研究无限维李代数是很自然的。例如，无限维李代数最重要的家族之一，被称为仿射李代数，来自弦理论中的对称性。比李代数本身更重要的是它们作为空间对称的方式，称为表征。李代数通常有很多表示，理解它们可以提供更多关于李代数的信息，甚至关于李代数产生的系统的物理。本研究计划的目标是研究仿射李代数的结构和泛化的表示，特别强调这些李代数的多变量类似物。特别是，我们期望给出几个这样的表示的第一个描述和分类，我们将使用内部对称性（称为自同构）来得出关于这些对象结构的结论。

Lie algebras are mathematical objects that describe the symmetries of certain physical systems.  Since many such systems have infinitely many independent symmetries, it is natural to study infinite-dimensional Lie algebras.  For example, one of the most important families of infinite-dimensional Lie algebras, called the affine Lie algebras, come from symmetries occurring in string theory.Even more important than the Lie algebras themselves are the ways in which they act as symmetries of spaces, called representations.  A Lie algebra typically has a great many representations, and understanding them can give more information about the Lie algebra, or even about the physics of the system from which the Lie algebra arises.The goal of this research program is to investigate the structure and representations of generalizations of the affine Lie algebras, with special emphasis on multivariable analogues of these Lie algebras.  In particular, we expect to give the first description and classification of several such representations, and we will use internal symmetries (called automorphisms) to draw conclusions about the structure of these objects.