Gabriel's Theorem for Locally Finite-Dimensional Representations of Infinite Quivers

TL;DR

This paper extends Gabriel's theorem to classify the representation types of locally finite-dimensional representations of infinite quivers, linking them to generalized ADE Dynkin diagrams.

Contribution

It establishes a characterization of infinite quivers whose representation categories have unique representation type, generalizing classical Gabriel's theorem.

Findings

Unique representation type corresponds to generalized ADE Dynkin diagrams.

Classification applies to infinite quivers like A_infinity and D_infinity.

Provides a criterion for the structure of representations of infinite quivers.

Abstract

We prove a version of Gabriel's theorem for locally finite-dimensional representations of infinite quivers. Specifically, we show that if $\Omega$ is any connected quiver, the category of locally finite-dimensional representations of $\Omega$ has unique representation type (meaning no two indecomposable representations have the same dimension vector) if and only if the underlying graph of $\Omega$ is a generalized ADE Dynkin diagram (i.e. one of $A_n, D_n, E_6, E_7, E_8, A_{\infty}, A_{\infty , \infty}$ or $D_\infty$). This result is companion to earlier work of the authors generalizing Gabriel's theorem to infinite quivers with different conditions.