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: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759328
• 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的联系的Artin代数的代表理论。我们的方法包括AR理论，倾斜理论和涵盖理论。 1）对于Artin代数，我们将描述某些特殊AR组件的形状；分类表示自由基具有较小的尼尔氏率的代数；根据其Ar-quievrs来表征表示代表性的；在遗传代数上构建所有预注射式倾斜模块，并表征集群倾斜的代数。 对于带有无限非零路径或无限dynkin的物种的绳子箭袋，我们将分类不可分配的代表并描述其AR组件。2）使用三角形类别的扩展子类别的封闭子类别，我们将在模块类别中独立地研究AR-AR-Becories类别，并确切的abelian，Abelian，Abelian，Trrian，Trtrian，Trrian Gianian类别。这将在具有无限的非零路径的结合箭量类别中产生几乎分裂序列的存在定理，而无限径的无限质量阵阵物种的存在。 3）我们将提供一些新的观点来研究代数的同源特性。我们将不会攻击有限全球维度的Artin代数类别的Ar-quiver的循环猜想，并建立具有有限奇异性类别的代数的有限维度猜想。我们将在基本代数的颤动中找到一个定向周期的标准，以支持一个半精简的无限射击尺寸模块。我们将不为具有自由基的零立方体代数的ARTIN代数和延伸猜想建立循环猜想。 4）我们应为二次单元代数的派生类别构建一个Galois覆盖物，以便对不可传代的复合物进行分类并描述其在字符串案例中的AR组件。我们将研究一个新类别，即通过有限维度表示的三角剖分子类别有限地呈现有限局部有限的箭袋的有限端表示的衍生类别的verdier商。 5）我们将表明，有限箭箭过物种的代表类别的派生类别的规范轨道类别是一个类别，并且对相应的群集代数分类。我们将构建B型无穷大和C无穷大的类别类别。6）给定A型无穷大或双重无穷大的群集类别，我们将对刚性的刚性子类别的标准感兴趣，最大程度是刚性的，并且对构造群集倾斜的子类别的方法具有群集类别。如果是有限类型的dynkin颤抖。