Mathematical Sciences: Representation Theory of Lie Algebras

恩莱特教授的数学研究计划 三个主要方面。首先是一项研究的延续 研究微分系统之间的联系 方程，某些称为行列式的仿射变量 变种，和李群的最高权重表示。 第二个目标是提出一个特殊函子理论 关于李代数的模范畴，它允许 Zuckerman和Vogan的函子方法推广到 域，包括Kac - Moody李代数。第三方面 是对Kac-Moody表示的么正性的研究 代数 这个项目与谎言的表征理论有关 群（及其表兄弟，李代数）， 挪威数学家Sophus Lie。一个基本的例子 李群是球体的旋转群，其中群 操作包括一个动作接着另一个动作。详细 关于这个群体的信息非常有助于解决 球对称的数学或物理问题 礼物其他的运动组也有其他的对称性。 一个更代数（相对于几何）的例子来源， 李群来自矩阵的乘法。本集团 对于给定大小的所有可逆真实的（或复）矩阵， 李群，就像它的任何子群一样， 以自然的方式描述。能去是令人向往的 在几何点和代数点之间来回 例如，考虑到许多方法， 球体的旋转组可以被实现为一组 可逆矩阵这就是表象理论。 关于给定群体的表示的事实倾向于存储一个 大量信息非常经济。转让本 信息，比如说在两种不同的跟踪方式之间， 一个群或代数的表示 恩莱特教授正在学习。尽管李反对 在经典意义上是有限维的，最近有 一直对他们的无限维 类似物，其调查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.