Representation theory of algebras and related topics

点击翻译按钮获取中文摘要

This project consists of four topics on the representation theory of artin algebras and related areas. 1) We shall study how to decide whether an artin algebra is of finite or infinite global dimension. For a finite dimensional algebra over an algebraically closed field, we shall approach this problem by studying the oriented cycles in the Gabriel quiver of the algebra. For an artin algebra, we shall approach this problem by considering its Cartan determinant. That is, we want to establish, at least for more special classes of algebras, the Cartan Determinant Conjecture which says that the global dimension is infinite whenever the Cartan determinant does not take value one.2) We shall study the Auslander-Reiten theory further in a general additive category. In case the category is Hom-finite and Krull-Schmidt, we shall try to find a formula which unifies the Auslander-Reiten duality for abelian categories and the Serre duality for triangulated categories. In case the category is 2-Calabi-Yau triangulated, we shall try to obtain an explicit description of its Auslander-Reiten components. 3) We want to study the derived category of an algebra, since it measures the complexity of the homological behavior of the algebra. For this purpose, we shall develop a covering technique for derived categories of locally bounded categories, and apply it to investigate the derived category of an algebra with radical squared-zero. Inspired from the characterization of tilted algebras, we shall be interested in characterizing algebras derived equivalent to a hereditary algebra by the existence of a complete "slice" in the Auslander-Reiten quiver of their derived category. 4) We shall apply our knowledge on the representations of infinite quivers to study cluster categories with an infinite cluster structure. It is particularly realistic for us to study the cluster categories of infinite Dynkin types, since we have shown that the Auslander-Reiten components of the derived category of the finitely presented representations of an infinite Dynkin quiver are all standard.