On locally semi-simple representations of quivers

TL;DR

This paper characterizes tame acyclic quivers by the property that all locally semi-simple representations have semi-simple endomorphism rings, solving a problem posed by Kac.

Contribution

It provides a complete characterization of tame quivers via the local semi-simplicity of certain representations, addressing a question by Kac.

Findings

Acyclic quiver is tame iff all representations with semi-simple endomorphism rings are locally semi-simple.

The paper establishes a necessary and sufficient condition linking tame type to local semi-simplicity.

It advances understanding of the structure of quiver representations and their endomorphism rings.

Abstract

In this paper, we solve a problem raised by V. Kac in \cite{Kac} on locally semi-simple quiver representations. Specifically, we show that an acyclic quiver $Q$ is of tame representation type if and only if every representation of $Q$ with a semi-simple ring of endomorphisms is locally semi-simple.