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 artin 代数表示论及相关领域的四个主题组成。 1）我们将研究如何判断一个artin代数的全局维数是有限维还是无限维。对于代数闭域上的有限维代数，我们将通过研究代数加布里埃尔颤动中的定向循环来解决这个问题。对于 artin 代数，我们将通过考虑其嘉当行列式来解决这个问题。也就是说，至少对于更特殊的代数类，我们想要建立嘉当行列式猜想，该猜想表明，只要嘉当行列式不取值一，全局维数就是无限的。 2）我们将在一般加性范畴中进一步研究Auslander-Reiten理论。如果范畴是 Hom-finite 和 Krull-Schmidt，我们将尝试找到一个公式来统一阿贝尔范畴的 Auslander-Reiten 对偶性和三角范畴的 Serre 对偶性。如果类别是 2-Calabi-Yau 三角剖分，我们将尝试获得其 Auslander-Reiten 分量的明确描述。 3）我们想要研究代数的派生范畴，因为它衡量代数同调行为的复杂性。为此，我们将开发一种局部有界范畴的派生范畴的覆盖技术，并将其应用于研究根式平方为零的代数的派生范畴。受倾斜代数表征的启发，我们将感兴趣的是通过其派生范畴的 Auslander-Reiten 箭袋中存在完整的“切片”来表征与遗传代数等价的派生代数。 4）我们将应用我们关于无限颤动表示的知识来研究具有无限簇结构的簇类别。对我们来说，研究无限 Dynkin 类型的簇范畴尤其现实，因为我们已经证明，无限 Dynkin 箭袋的有限呈现表示的派生范畴的 Auslander-Reiten 分量都是标准的。