Strongly irreducible operators and indecomposable representations of quivers on infinite-dimensional Hilbert spaces

TL;DR

This paper explores indecomposable representations of quivers on infinite-dimensional Hilbert spaces, utilizing strongly irreducible operators, and constructs examples with trivial endomorphism rings, linking operator theory with quiver representations.

Contribution

It introduces new methods to construct indecomposable Hilbert space representations of the Kronecker quiver with trivial endomorphism rings, connecting operator theory and quiver representations.

Findings

Constructed transitive indecomposable representations via perturbations of weighted shift operators.

Modified unbounded operators to produce transitive lattice representations.

Linked problems in operator theory to quiver representation theory.

Abstract

We study several classes of indecomposable representations of quivers on infinite-dimensional Hilbert spaces and their relation. Many examples are constructed using strongly irreducible operators. Some problems in operator theory are rephrased in terms of representations of quivers. We shall show two kinds of constructions of quite non-trivial indecomposable Hilbert representations of the Kronecker quiver such that their endomorphism rings are trivial, which are called transitive. One is a perturbation of a weighted shift operator by a rank-one operator. The other one is a modification of an unbounded operator used by Harrison,Radjavi and Rosenthal to provide a transitive lattice.