Indecomposable representations of quivers on infinite-dimensional Hilbert spaces

Abstract

We study indecomposable representations of quivers on separable infinite-dimensional Hilbert spaces by bounded operators. We consider a complement of Gabriel's theorem for these representations. Let $\Gamma$ be a finite, connected quiver. If its underlying undirected graph contains one of extended Dynkin diagrams $\tilde{A_n} (n \geq 0)$, $\tilde{D_n} (n \geq 4)$, $\tilde{E_6}$,$\tilde{E_7}$ and $\tilde{E_8}$, then there exists an indecomposable representation of $\Gamma$ on separable infinite-dimensional Hilbert spaces.