Moduli Techniques in Graded Ring Theory and Their Applications

分级环理论中的模技术及其应用

基本信息

批准号：
EP/M008460/1
负责人：
Susan Sierra
金额：
$ 37.51万
依托单位：
University of Edinburgh
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2015
资助国家：
英国
起止时间：
2015 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FM008460%2F1
关键词：
Moduli Techniques Graded Ring Theory

项目摘要

A ring is a mathematical structure that models many types of symmetry. Most rings encountered "in nature" are noncommutative: the order of operations matters. This project will investigate deep relationships between noncommutative ring theory and geometry.Rings are studied through their modules: objects that echo the symmetry encoded in the ring. The structure of a ring depends subtly and powerfully on the geometry of families of modules over that ring, and this connection has led to many advances. This project will explore this connection between the geometry of families of modules and the algebraic structure of rings in depth. I will extend current methods and develop new ones, and will apply my results to important unsolved algebraic problems. An example of the power of this connection between geometry and algebra is given by the famous Virasoro algebra. The Virasoro algebra is renowned in mathematics and physics. It may be viewed as a mathematical model of statistical mechanics, and so is of deep importance to physics, particularly conformal field theory. The Virasoro algebra is a Lie algebra, rather than a ring; it can be turned into a ring by forming its so-called universal enveloping algebra. Although the Virasoro algebra had been intensively studied for many years, important basic questions about its universal enveloping algebra remained unanswered. Specifically, for at least 25 years mathematicians had been asking if the enveloping algebra of the Virasoro algebra had the noetherian property. (Rings that are noetherian are relatively well-behaved; those that are not noetherian are more exotic.) In recent joint work with Walton, I applied geometry to solve this problem: the enveloping algebra of the Virasoro algebra is not noetherian. Our work shows the power of geometric techniques to address purely algebraic problems.One key method of our proof that the enveloping algebra of the Virasoro algebra is not noetherian was to construct a simpler model, called the canonical birational commutative factor. Because it is simpler, the model is easier to study; on the other hand, passing to the model loses a great deal of information. In this project, I will develop a general method, which will apply to many more rings than the enveloping algebra of the Virasoro algebra, to construct other canonical factors that contain more information but are still amendable to study. A general construction of more complex canonical factors will be a significant advance.Through the new techniques this project will develop, I will answer many important questions in ring theory. I will use geometry to get more information about the enveloping algebra of the Virasoro algebra. I will explore whether the noetherian property described above can be detected through geometry. I will apply geometric methods to a large class of rings, of which the enveloping algebra of the Virasoro is only one example: to universal enveloping algebras of graded infinite-dimensional Lie algebras. Through these methods, I will show these rings are not noetherian. These rings are famously intractable, and this problem is inaccessible without the new methods that I will bring to bear.

环是一种模拟多种类型的对称性的数学结构。大多数“自然界中”遇到的戒指是非交通性的：操作顺序很重要。该项目将研究非交流环理论与几何形状之间的深层关系。环通过其模块进行研究：与环中编码的对称性的对象。环的结构巧妙而有力地取决于该戒指上模块家族的几何形状，并且这种连接导致了许多进步。该项目将探讨模块家族的几何形状与环的代数结构之间的联系。我将扩展当前方法并开发新方法，并将结果应用于重要的未解决代数问题。著名的Virasoro代数给出了几何和代数之间这种联系的力量的一个例子。 Virasoro代数以数学和物理学而闻名。它可以被视为统计力学的数学模型，因此对物理学，尤其是保形场理论至关重要。 Virasoro代数是一个谎言代数，而不是环。它可以通过形成其所谓的通用包围代数来变成环。尽管Virasoro代数已经进行了多年的深入研究，但有关其普遍包围代数的重要基本问题仍未得到解答。具体而言，至少25年来数学家一直在询问Virasoro代数的代数是否具有Noetherian财产。（noe夫人的戒指表现相对良好；那些不是noetherian的戒指更具异国情调。）在最近与沃尔顿的联合合作中，我应用了几何来解决这个问题：Virasoro代数的包围代数不是Noetherian。我们的工作显示了几何技术解决纯粹代数问题的力量。我们证明了Virasoro代数的代数的一种关键方法，并不是noyetherian构建一个更简单的模型，称为规范的birational polication tocolmational polithational timpartivation因子。因为它更简单，所以模型更容易研究。另一方面，传递模型会丢失大量信息。在这个项目中，我将开发一种通用方法，该方法将适用于比Virasoro代数的代数更多的环，以构建其他包含更多信息但仍可以修改的规范因素。更复杂的规范因素的一般结构将是一个重大的进步。通过该项目将开发的新技术，我将回答RING理论中的许多重要问题。我将使用几何形状获取有关Virasoro代数的代数的更多信息。我将探讨是否可以通过几何形状检测到上述的Noetherian属性。我将应用几何方法对大类环，其中virasoro的包围代数只是一个示例：通用渐变的无限维二维代数代数代数代数代数代数。通过这些方法，我将表明这些环不是noe夫。这些戒指非常棘手，如果没有我将带来的新方法，这个问题是无法访问的。