Categories of generalized thread quivers

TL;DR

This paper investigates the structure and classification of indecomposable representations in categories derived from thread quivers, revealing new hereditary categories and detailed decompositions.

Contribution

It introduces a classification of indecomposable representations of thread quivers and constructs new hereditary abelian categories with these representations.

Findings

Indecomposable representations are induced from simpler quivers.

Complete classification of injective and projective indecomposables under mild conditions.

Construction of new hereditary abelian categories including all indecomposables.

Abstract

We study the representation category of thread quivers and their quotients. A thread quiver is a quiver in which some arrows have been replaced by totally ordered sets. Pointwise finite-dimensional (pwf) representations of such a thread quiver admit a Krull-Remak-Schmidt-Azumaya decomposition. We show that an indecomposable representation is induced from an indecomposable representation of a quiver obtained from the original quiver by replacing some of its arrows by a finite linear $\mathbb{A}_n$ quiver. We study injective and projective pwf indecomposable representations and we fully classify them when the quiver satisfies a mild condition. We give a characterization of the indecomposable pwf representations of certain categories whose representation theory has similar properties to finite type or tame type. We further construct new hereditary abelian categories, including a Serre subcategory of pwf representations that includes every indecomposable representation.