Mathematical Sciences: Sklyanin Algebras, and Graded Algebras

本文研究了一类新的分级代数，包括Sklyanin代数、正则代数和与量子群相关的代数。我们的目标是用几何术语来理解他们的表征理论。首席研究员将研究Sklyanin代数的有限维简单模。他将研究高维Sklyanin代数的性质，如正则性和诺etherian链条件。他还将从变形的角度研究它们的特性。本研究属于非交换环理论的一般范畴。项目中考虑的环在数学的许多领域都很有趣，包括代数几何。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它们之间的联系将有助于确定给定曲线的几何形状。

This research is concerned with the study of new classes of graded algebras including Sklyanin algebras, regular algebras and algebras related to quantum groups. The goal is to understand their representation theory in geometric terms. The principal investigator will study the finite dimensional simple modules of Sklyanin algebras. He will examine higher dimensional Sklyanin algebras for properties such as regularity and noetherian chain conditions. He will also examine their properties from a deformation perspective. This research is in the general area of noncommutative ring theory. The rings considered in the project are of interest in many areas of mathematics including algebraic geometry. 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 determining the geometry of a given curve.