Preprojective algebras, skew group algebras and Morita equivalences

TL;DR

This paper establishes a Morita equivalence between skew group algebras of preprojective algebras and generalized preprojective algebras derived from Cartan triples, linking algebraic structures with folding processes.

Contribution

It introduces a Morita equivalence connecting skew group algebras of preprojective algebras to generalized preprojective algebras via Cartan triples, extending understanding of algebraic folding.

Findings

Morita equivalence between skew group and generalized preprojective algebras

Isomorphism of ideal monoids compatible with Weyl group embeddings

Connection between folding process and algebraic structures

Abstract

Let $\mathbb{K}$ 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\"{o}er. 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.