Interaction between Representation Theory of Algebras and Cluster Theory

代数表示论与簇论的相互作用

• 批准号: RGPIN-2018-06107
• 负责人: Liu, Shiping
• 金额: $ 1.68万
• 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: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=713611
• 关键词: Interaction; between; Representation; Theory; Algebras

Our objective is to study the representation theory of artin algebras with connection to cluster theory. Our methodology consists of AR-theory, tilting theory and covering theory. 1) For artin algebras, we shall describe the shapes of some special AR-components; classify representation-finite algebras whose radical has a small nilpotency; characterize the representation-finiteness in terms of their AR-quievrs; construct all preprojective or preinjective tilting modules over a hereditary algebra and characterize cluster tilted algebras. For a string bound quiver with infinite nonzero paths or a species of an infinite Dynkin valued quiver, we shall classify the indecomposable representations and describe their AR-components. 2) Working with extension closed subcategories of triangulated categories, we shall unify the AR-theory studied independently in module categories over rings, exact categories, abelian categories and triangulated categories. This will yield existence theorems of almost split sequences in the representation category of a bound quiver with infinite non-zero paths and that of species of infinite valued quivers with no infinite path. 3) We shall provide some new points of view to study the homological properties of algebras. We shall attack No Loop Conjecture from the AR-quiver of the derived category of artin algebras of finite global dimension and establish Finitistic Dimension Conjecture for algebras with a finite singularity category. We shall find a criterion for an oriented cycle in the quiver of an elementary algebra to support a semisimple module of infinite projective dimension. We shall establish No Loop Conjecture for artin algebras with radical cubed zero and Extension Conjecture for elementary algebras with radical cubed zero. 4) We shall construct a Galois covering for the derived category of a quadratic monomial algebra in order to classify the indecomposable complexes and describe their AR-components in the string case. We shall study a new category, that is the Verdier quotient of the derived category of finitely presented representations of a strongly locally finite quiver by the triangulated subcategory of finite dimensional representations. 5) We shall show that the canonical orbit category of the derived category of the representation category of a species of a finite valued quiver is a cluster category, and it categorifies the corresponding cluster algebra. We shall construct cluster categories of types B infinity and C infinity. 6) Given a cluster category of type A infinity or A double infinity, we shall be interested in a criterion for a rigid subcategory to be maximal rigid and in a method to construct all the cluster tilting subcategories. 7) Given a cluster category, we shall characterize its rank in terms of some of its intrinsic properties and show that it is the classical cluster category associated with a Dynkin quiver if it is of finite type.

我们的目标是研究与聚类理论相关的 artin 代数的表示理论。我们的方法论包括AR理论、倾斜理论和覆盖理论。 1）对于artin代数，我们将描述一些特殊AR分量的形状；对根式幂零性小的表示有限代数进行分类；根据 AR-quievrs 来表征表示有限性；在遗传代数上构造所有预投影或预射倾斜模块并表征簇倾斜代数。 对于具有无限非零路径的弦约束箭袋或无限 Dynkin 值箭袋的种类，我们将对不可分解的表示进行分类并描述它们的 AR 组件。 2）利用三角范畴的扩展封闭子范畴，我们将在环、精确范畴、阿贝尔范畴和三角范畴的模块范畴中统一独立研究的AR理论。这将在具有无限非零路径的有界颤动的表示范畴和没有无限路径的无限值颤动的物种的表示范畴中产生几乎分裂序列的存在定理。 3）我们将为研究代数的同调性质提供一些新的观点。我们将从有限全局维数 artin 代数的派生范畴的 AR 箭袋攻击无循环猜想，并为具有有限奇点范畴的代数建立有限维猜想。我们将在初等代数的颤动中找到一个有向环的准则，以支持无限射影维数的半单模。我们将为带有根式零的艺术代数建立无环猜想，为带有根式零的初等代数建立可拓猜想。 4) 我们将为二次单项代数的派生范畴构造一个伽罗瓦覆盖，以便对不可分解复形进行分类并描述它们在弦情况下的 AR 分量。我们将研究一个新范畴，即由有限维表示的三角子范畴导出的强局部有限颤动的有限表示表示的 Verdier 商。 5) 我们将证明有限值箭袋物种的表示范畴的派生范畴的正则轨道范畴是一个簇范畴，并且它对相应的簇代数进行了分类。我们将构造 B 无穷大和 C 无穷大类型的簇类别。 6) 给定 A 无穷大或 A 双无穷大类型的簇类别，我们将对刚性子类别成为最大刚性的标准以及构造所有簇倾斜子类别的方法感兴趣。 7) 给定一个簇类别，我们将根据其一些内在属性来描述其等级，并证明它是与 Dynkin 箭袋相关的经典簇类别（如果它是有限类型）。