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

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