Gabriel's Theorem for Locally Finite-Dimensional 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 $Ω$ is any connected quiver, the category of locally finite-dimensional representations of $Ω$ has unique representation type (meaning no two indecomposable representations have the same dimension vector) if and only if the underlying graph of $Ω$ 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.