Interaction between Representation Theory of Algebras and Cluster Theory

我们的目标是研究与簇理论相关的阿廷代数的表示理论。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的AR-分支的形状，对根具有小幂零性的表示有限代数进行分类，用AR-quievr刻画表示有限性，构造遗传代数上的所有预投射或预内射倾斜模，并刻画簇倾斜代数. 对于一个具有无限非零路径的弦有界代数或一类无限Dynkin值代数，我们将对不可分解表示进行分类并描述它们的AR-分量。 2)利用三角范畴的扩张闭子范畴，我们将把独立研究的AR-理论统一到环上的模范畴、正合范畴、阿贝尔范畴和三角范畴中。这将产生几乎可裂序列的存在性定理的表示范畴的有界图与无限的非零路径和物种的无限值箭图没有无限的道路。 3)这将为研究代数的同调性质提供一些新的观点。本文从有限整体维数的artin代数的导出范畴的AR-代数出发，攻击了无圈猜想，并建立了有限奇异范畴代数的无圈维数猜想。我们将在初等代数的代数环中找到一个有向圈支持无限投射维数半单模的准则。本文建立了根为三次零的artin代数的无圈猜想和根为三次零的初等代数的扩张猜想。 4)为了对不可分解复形进行分类并描述它们在弦情形下的AR-分量，我们将构造二次单项式代数的导出范畴的伽罗瓦覆盖。我们将研究一个新的范畴，即强局部有限群的有限维表示的三角化子范畴的表示的导出范畴的Verdier商。 5)我们将证明有限值群的表示范畴的导出范畴的标准轨道范畴是一个簇范畴，并且它范畴化了相应的簇代数。我们将构造类型为B infinity和C infinity的簇范畴。 6)给定一个A型无穷大或A型双无穷大的簇范畴，我们感兴趣的是一个刚性子范畴是极大刚性的准则和一个构造所有簇倾斜子范畴的方法。 7)给定一个簇范畴，我们将根据它的一些内在性质来刻画它的秩，并证明如果它是有限型的，它就是与Dynkin簇相关联的经典簇范畴。

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.