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 $\Omega$ 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.

19 pages, 1 figure