IndecomposaЫe representations of quivers on infinite-dimensional Hilbert spaces

摘要

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 Г be a finite, connected quiver. If its underlying undirected graph contains one of extended Dynkin diagrams A_n (n__0), D_n (n __4), __6, __7 and __8, then there exists an indecomposable representation of Г on separable infinite-dimensional Hubert spaces.