Homological properties of quantum cluster algebras

量子簇代数的同调性质

• 批准号: 2436773
• 金额: --
• 依托单位: Lancaster University
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2020
• 资助国家: 英国
• 起止时间: 2020 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2436773
• 关键词: Homological; properties; quantum; cluster; algebras

This project aims to explore the different types of behaviour of a family of algebras from a homological and geometric perspective. The algebras in question, graded quantum cluster algebras, have a rich combinatorial structure but many questions about them from other points of view, such as their ring-theoretic properties, are not well understood. For example, recent work of Booker-Price on rank 3 cluster algebras has found a variety of possible behaviours for the growth of these graded algebras, and shows that these behaviours correspond to combinatorial properties of the cluster data. The first part of the project will explore this in more generality, so that when we have a better understanding of the range of behaviours, we will be able to identify families having - or not having - desirable homological properties such as being AS-Gorenstein or AS-regular. When we have one of these homological properties, we may then apply the machinery of noncommutative geometry. Moving into this realm of noncommutative geometry opens up the possibility of studying the representation theory of quantum cluster algebras - something that has not been done before. There are too many representations to study them individually but instead one can hope to show that they are parameterised by nice geometric objects, such as curves or surfaces in projective space.

这个项目的目的是从同调和几何的角度来探索代数家族的不同类型的行为。代数的问题，分次量子簇代数，有丰富的组合结构，但许多问题，他们从其他角度来看，如他们的环理论的性质，没有得到很好的理解。例如，最近的工作布克价格排名3集群代数已经发现了各种可能的行为增长这些分级代数，并表明这些行为对应的组合属性的集群数据。该项目的第一部分将更普遍地探索这一点，以便当我们更好地了解行为的范围时，我们将能够识别具有或不具有理想的同源特性的家族，例如AS-Gorenstein或AS-正则。当我们有了这些同调性质之一时，我们就可以应用非对易几何的机制。进入非对易几何的这个领域，为研究量子团簇代数的表示论开辟了可能性--这是以前没有做过的。有太多的表示法来单独研究它们，但相反，人们可以希望表明它们是由漂亮的几何对象参数化的，例如射影空间中的曲线或曲面。