Unbounded strongly irreducible operators and transitive representations of quivers on infinite-dimensional Hilbert spaces

TL;DR

This paper introduces unbounded strongly irreducible and transitive operators related to indecomposable Hilbert space representations of quivers, extending the theory of unbounded operators and characterizing transitive representations for extended Dynkin diagrams.

Contribution

It defines new classes of operators and explores their relation to transitive and indecomposable quiver representations on infinite-dimensional Hilbert spaces.

Findings

Transitive Hilbert representations exist for extended Dynkin diagrams but not for oriented cyclic quivers.

Transitive representations imply indecomposability, but not vice versa.

The theory generalizes unbounded operators through quiver representations.

Abstract

We introduce unbounded strongly irreducible operators and transitive operators. These operators are related to a certain class of indecomposable Hilbert representations of quivers on infinite-dimensional Hilbert spaces. We regard the theory of Hilbert representations of quivers is a generalization of the theory of unbounded operators. A non-zero Hilbert representation of a quiver is said to be transitive if the endomorphism algebra is trivial. If a Hilbert representation of a quiver is transitive, then it is indecomposable. But the converse is not true. Let $\Gamma$ be a quiver whose underlying undirected graph is an extended Dynkin diagram. Then there exists an infinite-dimensional transitive Hilbert representation of $\Gamma$ if and only if $\Gamma$ is not an oriented cyclic quiver.