Indecomposable representations of quivers on infinite-dimensional Hilbert spaces

TL;DR

This paper investigates indecomposable representations of quivers on infinite-dimensional Hilbert spaces, extending classical finite-dimensional results and demonstrating the existence of such representations for certain extended Dynkin diagrams.

Contribution

It extends Gabriel's theorem to infinite-dimensional Hilbert space representations, showing existence results for indecomposables on specific quivers.

Findings

Existence of indecomposable representations for quivers with extended Dynkin diagrams.

Extension of Gabriel's theorem to infinite-dimensional Hilbert spaces.

Identification of conditions under which indecomposables exist.

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.