Mathematical Sciences: Rings of Differential Operators and Enveloping Algebras

本研究将重点研究微分 代数簇上的算子及其与 半单李的包络代数中的本原理想 代数给定一条曲线，有一个有限维代数（a 微分算子环的某因子环 与曲线的每个奇异点相关联。一个普遍问题 是理解局部几何 在一个奇异点，和环的结构， 微分算子 本文的研究是在非交换环的一般范畴内进行的 理论 在这个项目中考虑的戒指是伟大的 对数学的许多部分感兴趣。 给定一条曲线， 这些环可以与曲线上的某些点相关联。 更好地了解这些环和这种联系将 在确定给定曲线的几何形状时是有用的。

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.