Skew group algebras of deformed preprojective algebras

TL;DR

This paper classifies indecomposable modules over skew group algebras of deformed preprojective algebras for abelian groups, introduces a reflection functor, and establishes Morita equivalence with a new quiver algebra.

Contribution

It provides a complete description of indecomposable modules over skew group algebras of deformed preprojective algebras and constructs a Morita equivalence with a new quiver algebra.

Findings

Classification of indecomposable modules for abelian G

Introduction of a reflection functor for module categories

Morita equivalence between skew group algebra and a new quiver algebra

Abstract

Suppose that $Q$ is a finite quiver and $G\subseteq \Aut(Q)$ is a finite group, $k$ is an algebraic closed field whose characteristic does not divide the order of $G$. For any algebra $\Lambda=kQ/{\mathcal {I}}$, $\mathcal {I}$ is an arbitrary ideal of path algebra $kQ$, we give all the indecomposable $\Lambda G$-modules from indecomposable $\Lambda$-modules when $G$ is abelian. In particular, we apply this result to the deformed preprojective algebra $\Pi_{Q}^{\lambda}$, and get a reflection functor for the module category of $\Pi_{Q}^{\lambda}G$. Furthermore, we construct a new quiver $Q_{G}$ and prove that $\Pi_{Q}^{\lambda}G$ is Morita equivalent to $\Pi_{Q_{G}}^{\eta}$ for some $\eta$.