Mathematical Sciences: Representation Theory of Lie Algebras

恩赖特教授的数学研究纲领主要有三个方面。第一个是继续研究微分方程组、某些称为行列式变元的仿射变种和李群的最高权表示之间的联系。第二个目标是给出李代数模范畴上的特殊函子理论，它将Zuckerman和Vogan的函子方法推广到包含Kac-Moody李代数的整环上。第三个方面是关于Kac-Moody代数表示的单元化的研究。这个项目与李群(以及它们的近亲李代数)的表示理论有关，李群以挪威数学家索菲斯·李的名字命名。李群的一个基本例子是球体的旋转群，其中群运算包括跟随一个运动接着另一个运动。关于这个群的详细信息对解决存在球对称的数学或物理问题非常有帮助。其他运动组捕捉到了其他类型的对称性。李群的例子的一个更代数的(相对于几何的)来源来自于矩阵的乘法。所有给定大小的可逆实(或复)矩阵组成的群是李群，就像它的任何可以用自然方式描述的子群一样。希望能够在几何和代数观点之间来回移动，例如考虑可以将球体的旋转群实现为一组可逆矩阵的多种方式。粗略地说，这就是表象理论。关于给定群体的陈述的事实往往会非常经济地存储大量信息。恩赖特教授正在研究的是这种信息的传递，比如在记录单个群或代数的表示的两种不同方式之间。虽然经典意义上的谎言对象是有限维的，但最近人们对它们的无限维类似物非常感兴趣，其研究也是该项目的一部分。

Professor Enright's program of mathematical research has three main aspects. The first is the continuation of a study investigating the connections among systems of differential equations, certain affine varieties called determinantal varieties, and highest weight representations for a Lie group. The second objective is to present a theory of special functors on categories of modules for a Lie algebra which will allow the extension of the functorial methods of Zuckerman and Vogan to a domain which includes Kac - Moody Lie algebras. The third aspect is a study of unitarizability for representations of Kac -Moody algebras. This project has to do with the representation theory of Lie groups (and of their cousins, Lie algebras), which bear the name of the Norwegian mathematician Sophus Lie. One basic example of a Lie group is the group of rotations of a sphere, where the group operation consists of following one motion by another. Detailed information concerning this group is very helpful in solving mathematical or physical problems in which spherical symmetry is present. Other groups of motions capture other kinds of symmetry. A more algebraic (as opposed to geometric) source of examples of Lie groups comes from the multiplication of matrices. The group of all invertible real (or complex) matrices of a given size is a Lie group, as is just about any subgroup thereof that can be described in a natural manner. It is desirable to be able to go back and forth between the geometric and algebraic points of view, for instance to consider the numerous ways in which the rotation group of the sphere can be realized as a group of invertible matrices. This, roughly, is representation theory. Facts about the representations of a given group tend to store a lot of information very economically. Transfer of this information, say between two different ways of keeping track of the representations of a single group or algebra, is something Professor Enright is involved in studying. Although Lie objects in the classical sense are finite-dimensional, there has lately been a great deal of interest in their infinite-dimensional analogues, whose investigation is also part of this project.