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=P(Q,W). By a result of Reiten and Riedtmann, the quiver QG 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 WG on QG such that η(ΛG)η≅P(QG,WG). 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.