Mathematical Sciences: Representation Theory of Lie Algebras

数学科学：李代数表示论

• 批准号: 8902425
• 负责人: Thomas Enright
• 金额: $ 15.78万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1989
• 资助国家: 美国
• 起止时间: 1989-06-15 至 1993-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8902425&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Representation; Theory; Lie

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.

恩莱特教授的数学研究计划主要包括三个方面。第一个是继续研究微分方程组、某些称为行列式簇的仿射簇以及李群的最高权重表示之间的联系。 第二个目标是提出一种关于李代数模类别的特殊函子理论，该理论将允许将 Zuckerman 和 Vogan 的函子方法扩展到包括 Kac - Moody Lie 代数的域。第三个方面是 Kac-Moody 代数表示的幺正性研究。 该项目与李群（及其近亲李代数）的表示论有关，该表示论以挪威数学家 Sophus Lie 的名字命名。李群的一个基本示例是球体的旋转群，其中群运算由一个运动跟随另一个运动组成。有关该群的详细信息对于解决存在球对称性的数学或物理问题非常有帮助。其他组的运动捕捉其他类型的对称性。 李群例子的一个更代数（而不是几何）的来源来自矩阵的乘法。给定大小的所有可逆实数（或复数）矩阵的群是李群，几乎可以以自然方式描述的任何子群也是如此。我们希望能够在几何和代数的观点之间来回切换，例如考虑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，这就是表示论。 有关给定群体的表征的事实往往会非常经济地存储大量信息。恩莱特教授正在研究这种信息的传递，例如在跟踪单个群或代数表示的两种不同方式之间的传递。尽管经典意义上的李物体是有限维的，但最近人们对它们的无限维类似物产生了很大的兴趣，其研究也是该项目的一部分。