Gabriel’s Theorem for Infinite Quivers

We prove a version of Gabriel’s theorem for (possibly infinite dimensional) representations of infinite quivers. More precisely, we show that the representation theory of a quiver \(\varvec{\Omega }\) is of unique type (each dimension vector has at most one associated indecomposable) and infinite Krull-Schmidt (every, possibly infinite dimensional, representation is a direct sum of indecomposables) if and only if \(\varvec{\Omega }\) is eventually outward and of generalized ADE Dynkin type (\(\varvec{A_n}\), \(\varvec{D_n}\), \(\varvec{E_6}\), \(\varvec{E_7}\), \(\varvec{E_8}\), \(\varvec{A_\infty }\), \(\varvec{A_{\infty , \infty }}\), or \(\varvec{D_\infty }\)). Furthermore we define an analog of the Euler-Tits form on the space of eventually constant infinite roots and show that a quiver is of generalized ADE Dynkin type if and only if this form is positive definite. In this case the indecomposables are all locally finite-dimensional and eventually constant and correspond bijectively to the positive roots (i.e. those of length \(\varvec{1}\)).