Mathematical Sciences: Rings of Differential Operators and Enveloping Algebras

本文主要研究代数簇上的微分算子及其与半单李代数包络代数中本原理想的关系。给定一条曲线，该曲线的每个奇点都有一个有限维代数(微分算子环的某个因子环)。一个普遍的问题是理解奇点上的局部几何与微分算子环的结构之间的精确关系。本研究属于非对易环论的一般领域。这个项目中考虑的环对数学的许多部分都非常感兴趣。给定一条曲线，其中一个环可以与曲线上的某些点相关联。更好地理解这些环和这种联系将有助于确定给定曲线的几何形状。

This research will focus on the study of differential operators on algebraic varieties and the connections with primitive ideals in the enveloping algebra of a semi-simple Lie algebra. Given a curve, there is a finite dimensional algebra (a certain factor ring of the ring of differential operators) associated to each singular point of the curve. A general problem is to understand the precise relation between the local geometry at a singular point, and the structure of the ring of differential operators. This research is in the general area of noncommutative ring theory. The rings considered in this project are of great interest in many parts of mathematics. Given a curve, one of these rings can be associated with certain points on the curve. A better understanding of these rings and this association will be useful in determing the geometry of a given curve.