Mathematical Sciences: Representation Theory of Lie Algebras

恩莱特教授的数学研究计划主要有三个方面。第一个是继续研究微分方程系统之间的联系，某些被称为行列式的仿射变量，以及李群的最高权表示。第二个目标是提出一个关于李代数模的范畴上的特殊函子的理论，它允许将Zuckerman和Vogan的函子方法推广到包含Kac - Moody李代数的域。第三个方面是对Kac -Moody代数表示的一元性的研究。这个项目与李群的表示理论（以及它们的近亲，李代数）有关，它以挪威数学家Sophus Lie的名字命名。李群的一个基本例子是球体的旋转群，其中群运算包括一个运动跟随另一个运动。关于这一群体的详细信息对于解决球面对称存在的数学或物理问题非常有帮助。其他组的运动捕捉到其他种类的对称性。李群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。给定大小的所有可逆实（或复）矩阵的群是李群，就像它的任何子群一样，可以用自然的方式来描述。我们希望能够在几何和代数的观点之间来回转换，例如考虑球体的旋转群可以被实现为一组可逆矩阵的许多方法。粗略地说，这就是表征理论。关于特定群体表征的事实倾向于非常经济地存储大量信息。这种信息的传递，比如在两种不同的方式之间跟踪单个组或代数的表示，是Enright教授参与研究的内容。虽然经典意义上的Lie对象是有限维的，但最近人们对它们的无限维类似物产生了极大的兴趣，对它们的研究也是本项目的一部分。

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.