Skew group algebras of Jacobian algebras

Abstract

For a quiver with potential $(Q,W)$ with an action of a finite cyclic group $G$, we study the skew group algebra $ΛG$ of the Jacobian algebra $Λ= \mathcal P(Q, W)$. By a result of Reiten and Riedtmann, the quiver $Q_G$ of a basic algebra $η( ΛG) η$ Morita equivalent to $ΛG$ is known. Under some assumptions on the action of $G$, we explicitly construct a potential $W_G$ on $Q_G$ such that $η(ΛG) η\cong \mathcal P(Q_G , W_G)$. The original quiver with potential can then be recovered by the skew group algebra construction with a natural action of the dual group of $G$. If $Λ$ is self-injective, then $ΛG$ is as well, and we investigate this case. Motivated by Herschend and Iyama's characterisation of 2-representation finite algebras, we study how cuts on $(Q,W)$ behave with respect to our construction.