Skew group algebras of deformed preprojective algebras

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 $\eta$.

20 pages