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 $\Lambda G$ of the Jacobian algebra $\Lambda =\mathcal{P}(Q,W)$. By a result of Reiten and Riedtmann, the quiver $Q_G$ of a basic algebra $\eta (\Lambda G)\eta$ Morita equivalent to $\Lambda G$ is known. Under some assumptions on the action of $G$, 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 $\Lambda$ is self-injective, then $\Lambda G$ 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.

34 pages, comments welcome. Final version, to appear in Journal of Algebra