Preprojective algebras, skew group algebras and Morita equivalences

Let K be a field of characteristic p and G be a cyclic p-group which acts on a finite acyclic quiver Q. The folding process associates a Cartan triple to the action. We establish a Morita equivalence between the skew group algebra of the preprojective algebra of Q and the generalized preprojective algebra associated to the Cartan triple in the sense of Geiss, Leclerc and Schr\"oer. The Morita equivalence induces an isomorphism between certain ideal monoids of these preprojective algebras, which is compatible with the embedding of Weyl groups appearing in the folding process.