Partial actions of groups on quivers and path algebras
Wagner Cortes, Eduardo N. Marcos
TL;DR
This paper develops the theory of partial group actions on quivers and their path algebras, introducing partial actions on quivers and the enveloping (global) actions that realize them. It further generalizes to algebras by subalgebras, defining globalization and enveloping actions in this setting. The main result shows that any partial action on a quiver has an enveloping action, and that the corresponding enveloping action on the path algebra $KQ$ serves as the enveloping action for the induced partial action on $K\Gamma$, with the enveloping structures on vertices and arrows determined by $G$-orbits. These results unify partial actions on quivers with their algebraic counterparts and provide a concrete construction for enveloping actions on path algebras, along with illustrative examples and distinctions from Exel–Dokuchaev style partial actions.
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 $Γ$, there exists another quiver, $Γ'$ with a full $G$-action. This is an enveloping action of the partial action of $G$ on $Γ$. 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.