Skew group algebras of Jacobian algebras

TL;DR

This paper investigates skew group algebras of Jacobian algebras derived from quivers with potential under cyclic group actions, explicitly constructing potentials on the resulting quivers and exploring properties like self-injectivity and 2-representation finiteness.

Contribution

It provides an explicit construction of potentials on quivers after skew group algebra formation, extending the understanding of their structure and properties under group actions.

Findings

Explicit potential construction on the new quiver $Q_G$

Preservation of self-injectivity in the skew group algebra

Insights into cuts and 2-representation finiteness under the construction

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.