Skew group algebras of deformed preprojective algebras

Suppose that Q is a finite quiver and G subset of 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/I, where I is an arbitrary ideal of path algebra kQ, we give all the indecomposable AG-modules from indecomposable Lambda-modules when G is abelian. In particular, we apply this result to the deformed preprojective algebra Pi(lambda)(Q), and get a reflection functor for the module category of Pi(lambda)(Q)G Furthermore, we construct a new quiver Q(G) and prove that Pi(lambda)(Q)G is Morita equivalent to Pi(eta)(QG) for some eta. (C) 2011 Elsevier Inc. All rights reserved.