On a classification problem for a quiver of type $\widetilde{A}_{3}$

TL;DR

The paper addresses the classification of indecomposable representations of the affine quiver $\widetilde{A}_{3}$ by rephrasing the problem as classifying indecomposables for the quiver $\mathcal{S}$ of type $\widetilde{A}_{3}$, i.e., four linear maps between four finite-dimensional spaces. It introduces a matrix-based reduction that enables induction on the dimension of the first space, and accompanies this with graphical invariants to uniquely identify indecomposables, plus an explicit reconstruction algorithm from these invariants. It provides a complete description of endomorphism rings for all indecomposables, showing distinct ring structures depending on the type (e.g., $k[t]/(p^{s}(t))$ or $k[t]/(t^{n})$ or $k$). Furthermore, it demonstrates that the classical Kronecker problem and its contragredient version embed as subproblems within this framework, enabling transfer of the classification via simple identifications, and it connects these results to broader subproblems in representation theory with an elementary, self-contained approach.

Abstract

We present a new solution to the classification problem for the category of representations of a quiver of type $\widetilde{A}_{3}$. Our approach uses linear algebra techniques which lead us to a reduction that allows to use induction. As an application, the solution to the classical Kronecker problem and its contragredient version are obtained in an elementary way. We also describe the endomorphism rings for the indecomposable representations and an algorithm that shows how to reconstruct their matrix form from some graphic invariants.

Figures (3)

• Figure 1: Indecomposable representations of the quiver $\mathcal{S}$.
• Figure 2: Starting from $\mathcal{S}$ and taking $f_{\beta}$ and $f_{\delta}$ as identities, we can identify $V_{4}$, $V_{3}$, and $V_{2}$ obtaining the Kronecker quiver
• Figure 3: Starting from $\mathcal{S}$ and taking $f_{\beta}$ and $f_{\gamma}$ as identities, we can identify $V_{4}$ with $V_{1}$, and $V_{3}$ with $V_{2}$, obtaining the contragredient Kronecker quiver

Theorems & Definitions (12)

• Theorem 1.1
• Proposition 2.1
• proof
• proof : Proof of Theorem \ref{['the:indecomposable2x2']}
• Remark 1
• Remark 2
• Remark 3
• Theorem 3.1
• proof
• Corollary 4.1