Homological properties of quantum cluster algebras

这个项目旨在从同调和几何的角度探索代数家族的不同类型的行为。所讨论的代数，即梯度量子簇代数，具有丰富的组合结构，但从其他角度对它们的许多问题，如它们的环论性质，还没有得到很好的理解。例如，Booker-Price最近对3阶聚类代数的研究发现了这些分级代数生长的各种可能行为，并表明这些行为对应于聚类数据的组合属性。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 -regular。当我们有这些同调性质之一时，我们就可以应用非交换几何的机制。进入非交换几何的这个领域，开启了研究量子簇代数的表示理论的可能性——这是以前没有做过的事情。有太多的表征来单独研究它们，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，比如投影空间中的曲线或曲面。

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.