Quivers with potentials and actions of finite abelian groups

TL;DR

This paper explores how finite abelian groups acting on quivers with potentials induce Morita equivalences between associated Ginzburg dg algebras, providing explicit constructions and functorial relationships between their cluster categories.

Contribution

It explicitly constructs the potential $W_G$ on the quotient quiver and details the Morita equivalence between the Ginzburg dg algebras, extending understanding of group actions on quivers with potentials.

Findings

Explicit construction of the potential $W_G$ on the quotient quiver.

Detailed Morita equivalence between the Ginzburg dg algebras.

Functorial relations between the associated cluster categories.

Abstract

Let $G$ be a finite abelian group acting on a path algebra $kQ$ by permuting the vertices and preserving the arrowspans. Let $W$ be a potential on the quiver $Q$ which is fixed by the action. We study the skew group dg algebra $\Gamma_{Q, W}G$ of the Ginzburg dg algebra of $(Q, W)$. It is known that $\Gamma_{Q, W}G$ is Morita equivalent to another Ginzburg dg algebra $\Gamma_{Q_G, W_G}$, whose quiver $Q_G$ was constructed by Demonet. In this article we give an explicit construction of the potential $W_G$ as a linear combination of cycles in $Q_G$, and write the Morita equivalence explicitly. As a corollary, we obtain functors between the cluster categories corresponding to the two quivers with potentials.