Properness of moment maps for representations of quivers

TL;DR

This paper establishes a precise criterion linking the properness of moment maps in quiver representations to the acyclicity of the quiver, providing a clear geometric condition.

Contribution

It provides a necessary and sufficient condition for the properness of moment maps in quiver representations based on quiver acyclicity.

Findings

Properness of moment maps iff quiver is acyclic

Characterization of moment map properness in terms of quiver structure

Clear geometric criterion for moment map properness

Abstract

In this note, we give a necessary and sufficient condition for the properness of moment maps for representations of quivers. We show that the moment map for the representations of a quiver is proper if and only if the quiver is acyclic, that is, it does not contain any oriented cycle.