Indecomposable representations of quivers on infinite-dimensional Hilbert spaces

We study indecomposable representations of quivers on separable infinite-dimensional Hilbert spaces by bounded operators. We exhibit several concrete examples and investigate duality theorem between reflection functors. We also show a complement of Gabriel's theorem. Let Γ be a finite, connected quiver. If its underlying undirected graph contains one of extended Dynkin diagrams (n⩾0), (n⩾4), , and , then there exists an indecomposable representation of Γ on separable infinite-dimensional Hilbert spaces.