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 ˜An (n 0), ˜Dn (n 4), ˜E6, ˜E7 and ˜E8, then there exists an indecomposable representation of Γ on separable infinite-dimensional Hilbert spaces. © 2008 Elsevier Inc. All rights reserved.