Skew group algebras of Jacobian algebras

For a quiver with potential $(Q,W)$ with an action of a finite cyclic group $G$, we study the skew group algebra $\mathrm{\Lambda }G$ of the Jacobian algebra $\mathrm{\Lambda }=\mathcal{P}(Q,W)$. By a result of Reiten and Riedtmann, the quiver $Q_G$ of a basic algebra $\eta (\mathrm{\Lambda }G)\eta$ Morita equivalent to $\Lambda G$isknown\mathrm{.}Undersomeassumptionsontheactionof$G$, we explicitly construct a potential $W_G$ on $Q_G$ such that $\eta(\Lambda G) \eta\cong \mathcal P(Q_G , W_G)$. 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 $\mathrm{\Lambda }$ is self-injective, then $\mathrm{\Lambda }G$ 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.