Representation theory of algebras and related topics

代数表示论及相关主题

• 批准号: 172797-2013
• 负责人: Liu, Shiping
• 金额: $ 1.09万
• 依托单位: Université de Sherbrooke
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=569621
• 关键词: Representation; theory; algebras; related; topics

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.

该项目由关于阿丁代数表示理论及相关领域的四个主题组成。 1)我们将研究如何判定一个Artin代数是有限维的还是无限维的。对于代数闭域上的有限维代数，我们将通过研究代数的Gabriel环中的定向圈来解决这个问题。对于一个阿丁代数，我们将通过考虑它的嘉当行列式来处理这个问题。也就是说，我们要建立，至少对于更特殊的代数类，嘉当行列式猜想说，当嘉当行列式不取值1时，整体维数是无限的。 2)我们将在一般的加法范畴中进一步研究Auslander-Reiten理论。当范畴是Hom-finite和Krull-Schmidt范畴时，我们试图找到一个公式，它统一了阿贝尔范畴的Auslander-Reiten对偶和三角范畴的Serre对偶。如果范畴是2-Calabi-Yau三角化的，我们将试图得到它的Auslander-Reiten分支的一个明确的描述。 3)我们想研究代数的导出范畴，因为它度量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导范畴的覆盖技巧，并应用它来研究根为平方零的代数的导范畴。从倾斜代数的特征的启发，我们将感兴趣的特征代数导出等价于遗传代数的存在一个完整的“切片”在Auslander-Reiten的代数导出类别。 4)我们将应用我们的知识表示的无限箭研究集群范畴的无限集群结构。研究无限Dynkin型的簇范畴是特别现实的，因为我们已经证明了无限Dynkin型表示的导出范畴的Auslander-Reiten分支都是标准的.