Interaction between Representation Theory of Algebras and Cluster Theory

我们的目标是研究与聚类理论有联系的代数的表示理论。我们的方法包括ar理论、倾斜理论和覆盖理论。******1)对于n代数，我们将描述一些特殊ar分量的形状；分类根有小幂零的表示-有限代数表征表征的有限性在他们的ar颤动；构造遗传代数上的所有预投影或预内射倾斜模，并刻画聚类倾斜代数。对于具有无限非零路径的弦界颤振或无限Dynkin值颤振的一种，我们将对不可分解表示进行分类并描述它们的ar分量。******2)结合三角化范畴的可拓闭子范畴，将环上模范畴、精确范畴、阿贝尔范畴和三角化范畴中独立研究的ar理论统一起来。在具有无限非零路径的有界颤振的表示范畴和无无限路径的无限值颤振的表示范畴中，给出了几乎分裂序列的存在定理。******3)我们将提供一些新的观点来研究代数的同调性质。从有限整体维数的代数的派生范畴的ar颤振中攻击无环猜想，建立有限奇异范畴代数的有限维猜想。我们将在初等代数颤振中找到一个有向环的判据，以支持无限射影维数的半单模。建立了根次为零的代数的无环猜想和根次为零的初等代数的可拓猜想。******4)我们将构造二次一元代数的派生范畴的伽罗瓦覆盖，以便对弦情况下的不可分解复形进行分类并描述它们的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颤振有限表示范畴的Verdier商。******5)我们将证明有限值颤振的一种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。构造B无穷型和C无穷型的聚类范畴。******6)给定a无穷大或a双无穷大的聚类范畴，我们将研究刚性子范畴为极大刚性的准则和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表征它的秩，并证明如果它是有限型的，它就是与Dynkin颤振相关的经典簇范畴。

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.