Partial actions of groups on quivers and path algebras

TL;DR

This paper introduces the concept of partial group actions on quivers and path algebras, showing that such partial actions can be extended to full actions via enveloping quivers and algebras.

Contribution

It establishes the existence of enveloping actions for partial group actions on quivers and path algebras, extending the theory of group actions in algebraic structures.

Findings

Existence of enveloping quivers with full G-actions for any partial action

Partial actions on algebras can be extended to enveloping actions

Any partial action on a path algebra induced by a quiver action has an enveloping action

Abstract

In this article, we introduce the concept of partial actions of a group $G$ on quivers and demonstrate that for any given partial action of G on a quiver $\Gamma$, there exists another quiver, $\Gamma'$ with a full $G$-action. This is an enveloping action of the partial action of $G$ on $\Gamma$. We also introduce partial actions of groups on algebras by subalgebras instead of ideals and we define enveloping actions in this case. We show that any partial action of a group on a path algebra that is induced by a partial action on a quiver has an enveloping action.