Representation theory of algebras and related topics

这个项目包括四个关于Artin代数的表示理论和相关领域的主题。 1)我们将研究如何判定一个Artin代数是有限整体维还是无限整体维。对于代数闭域上的有限维代数，我们将通过研究该代数的Gabriel箭图中的有向圈来解决这个问题。对于Artin代数，我们将通过考虑它的Cartan行列式来解决这个问题。也就是说，我们想要建立，至少对于更特殊的代数类，Cartan行列式猜想，当Cartan行列式不取1的值时，全局维度是无穷的。 2)我们将在一般加法范畴中进一步研究Auslander-Reiten理论。在范畴是Hom-有限且Krull-Schmidt的情况下，我们将试图找到一个公式来统一阿贝尔范畴的Auslander-Reiten对偶和三角范畴的Serre对偶。如果范畴是2-Calabi-Yau三角剖分，我们将尝试获得其Auslander-Reiten分量的显式描述。 3)我们要研究代数的派生范畴，因为它度量了代数的同调行为的复杂性。为此，我们将发展局部有界范畴的派生范畴的覆盖技巧，并将其应用于研究具有根平方为零的代数的派生范畴。受倾斜代数刻画的启发，我们将感兴趣地刻画通过在其派生范畴的Auslander-Reiten箭图中存在完全“切片”来等价于遗传代数的派生代数。 4)我们将利用我们关于无限箭图表示的知识来研究具有无限簇结构的簇范畴。研究无限动态箭图的簇范畴对我们来说是特别现实的，因为我们已经证明了无限动态箭图的有限表示的派生范畴的Auslander-Reiten分支都是标准的。

This project consists of four topics on the representation theory of artin algebras and related areas. 1) We shall study how to decide whether an artin algebra is of finite or infinite global dimension. For a finite dimensional algebra over an algebraically closed field, we shall approach this problem by studying the oriented cycles in the Gabriel quiver of the algebra. For an artin algebra, we shall approach this problem by considering its Cartan determinant. That is, we want to establish, at least for more special classes of algebras, the Cartan Determinant Conjecture which says that the global dimension is infinite whenever the Cartan determinant does not take value one. 2) We shall study the Auslander-Reiten theory further in a general additive category. In case the category is Hom-finite and Krull-Schmidt, we shall try to find a formula which unifies the Auslander-Reiten duality for abelian categories and the Serre duality for triangulated categories. In case the category is 2-Calabi-Yau triangulated, we shall try to obtain an explicit description of its Auslander-Reiten components. 3) We want to study the derived category of an algebra, since it measures the complexity of the homological behavior of the algebra. For this purpose, we shall develop a covering technique for derived categories of locally bounded categories, and apply it to investigate the derived category of an algebra with radical squared-zero. Inspired from the characterization of tilted algebras, we shall be interested in characterizing algebras derived equivalent to a hereditary algebra by the existence of a complete "slice" in the Auslander-Reiten quiver of their derived category. 4) We shall apply our knowledge on the representations of infinite quivers to study cluster categories with an infinite cluster structure. It is particularly realistic for us to study the cluster categories of infinite Dynkin types, since we have shown that the Auslander-Reiten components of the derived category of the finitely presented representations of an infinite Dynkin quiver are all standard.