Categories of generalized thread quivers

TL;DR

This work extends quiver representation theory by introducing thread quivers $(Q,\mathcal{P})$ and studying their pointwise finite-dimensional representations via a two-stage decomposition into noise and noise-free parts. It develops a localization framework with valid partitions, completions, and induced representations from finite subthreadings to classify indecomposables and descends to a second, more precise decomposition theorem. Under left/right $Q$-boundedness it fully classifies indecomposable projectives/injectives and establishes hereditary properties for quasi noise-free representations, enabling new hereditary categories and connections to special biserial, gentle, and string categories. The paper also defines and analyzes new representation types (virtually finite/tame and essentially finite/tame), showing how threading can convert finite-type cases into virtually tame ones and providing a versatile toolkit for constructing and classifying representations of complex posets.

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 A thread quiver is a quiver in which some arrows have been replaced by totally ordered sets.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.

Figures (3)

• Figure 1: An illustration of $\alpha, \beta$ and $p.$
• Figure 2: The blue region on the left corresponds to the interval $(x,y)\subset (0,4)=\mathcal{P}_\alpha$. For our consideration, the red region on the right does not correspond to an interval.
• Figure 3: Table of functors. When $\mathcal{I}$ is $\mathfrak{P}$-complete, the quotient arrows $\mathcal{A}\twoheadrightarrow\overline{\mathcal{A}}^{\mathfrak{P}}$ and $\mathcal{A}(\mathcal{M})\twoheadrightarrow\overline{\mathcal{A}}^{\mathfrak{P}}(\mathcal{M})$ become identity arrows and the diagrams collapse to the lower triangles.

Theorems & Definitions (99)

• Theorem A: Theorems \ref{['thm:noise and noise free decomposition']} and \ref{['thm:second decomposition theorem']}
• Theorem B: Theorem \ref{['ThmAllProj']}
• Theorem C: Theorem \ref{['thm:quasi noise free is hereditary']} and Propositions \ref{['prop:no infinite paths and interval finite implies hereditary']} and \ref{['prop:fp_hereditary']}
• Definition 1.1
• Example 1.2
• Example 1.3
• Example 1.4
• Definition 1.5
• Definition 1.6
• Example 1.7