Representation theory of strongly locally finite quivers.

Abstract

this paper deals with the representation theory of strongly, locally finite quivers. We first study some properties of the finitely presented or co-presented representations, and then construct in the category of locally finite-dimensional representations some almost split sequences which start with a finitely co-presented representation and end with a finitely presented representation. Furthermore, we obtain a general description of the shapes of the Auslander–Reiten components of the category of finitely presented representations and prove that the number of regular Auslander–Reiten components is infinite if and only if the quiver is not of finite or infinite Dynkin type. In the infinite Dynkin case, we shall give a complete list of the indecomposable representations and an explicit description of the Auslander–Reiten components. Finally, we apply these results to study the Auslander–Reiten theory in the derived category of bounded complexes of finitely presented representations.