Interaction between Representation Theory of Algebras and Cluster Theory

我们的目的是研究与簇论相联系的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的AR-分支的形状；分类根具有小幂零的表示有限代数；用它们的AR-平凡刻画表示有限性；构造遗传代数上的所有预投射或预内射倾斜模；刻画簇倾斜代数。对于一个具有无限非零路的弦有界箭图或一个无限动态值箭图，我们将对其不可分解表示进行分类并刻画其AR-分量。*2)利用三角范畴的扩张闭子范畴，我们将统一在环上的模范畴、正则范畴、交换范畴和三角范畴上独立研究的AR-理论。这将产生具有无限非零路的有界箭图的表示范畴中的几乎分裂序列的存在性定理，以及没有无限路径的无穷值箭图的表示范畴的几乎分裂序列的存在性定理。*3)我们将为研究代数的同调性质提供一些新的观点。我们将从有限整体维Artin代数的派生范畴的AR-箭图出发来攻击No Loop猜想，并建立有限奇性范畴代数的初值维度猜想。我们将在初等代数箭图中找到一个判定有向圈的准则，以支持无限投射维半单模。我们将建立根为零的Artin代数的无环猜想和根为零的初等代数的扩张猜想。*4)我们将为二次单项代数的派生范畴构造Galois覆盖，以便对不可分解复形进行分类，并在弦情形下刻画它们的AR-分量。我们将研究一个新的范畴，即由有限维表示的三角化子范畴导出的强局部有限箭图的有限表示范畴的Verdier商。*5)我们将证明有限值箭图物种的表示范畴的派生范畴的正则轨道范畴是簇范畴，并且它对相应的簇代数进行范畴化。我们将构造B型无穷大和C型无穷大的簇范畴。*6)给定A型无穷或A双无穷大的簇范畴，我们将感兴趣的是刚性子范畴是极大刚性的一个判据以及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刻画它的等级，并证明如果它是有限类型的，则它是与动态箭图相关的经典簇范畴。

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.