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.

该项目包括关于Artin代数及相关领域的代表理论的四个主题。 1）我们将研究如何确定Artin代数是有限的还是无限的全球维度。对于在代数封闭场上的有限维代数，我们将通过研究代数的Gabriel颤动中的定向循环来解决此问题。对于Artin代数，我们将通过考虑其cartan的决定因素来解决此问题。也就是说，我们希望至少要建立更多特殊类别的代数，即cartan的决定因素猜想，即每当cartan决定因素不具有价值时，全球维度是无限的。 2）我们将在一般添加剂类别中进一步研究Auslander-Reiten理论。如果该类别是Hom-Finite和Krull-Schmidt，我们将尝试找到一个公式，该公式将Abelian类别的Auslander-Reiten双重性和三角形类别的Serre二元性统一。如果该类别是2-卡拉比三角剖分，我们将尝试获取其Auslander-Reiten组件的明确描述。 3）我们想研究代数的派生类别，因为它衡量了代数的同源行为的复杂性。为此，我们将针对本地界限类别的派生类别开发覆盖技术，并将其应用于以激进平方为零的代数的派生类别。受到倾斜代数的特征的启发，我们将有兴趣通过在其派生类别的Auslander-Reiten Quiver中存在一个完整的“切片”来表征相当于世袭代数的代数。 4）我们将应用有关无限颤动的表示的知识来研究具有无限集群结构的簇类别。对于我们来说，研究无限Dynkin类型的集群类别对我们来说尤为现实，因为我们已经表明，无限dynkin颤抖的有限呈现表示的衍生类别的Auslander-Reiten组件都是标准配置的。