Indecomposable Representations for Extended Dynkin Quivers

We describe a method for an explicit determination of indecomposable preprojective and preinjective representations for extended Dynkin quivers Γ over an arbitrary field K by vector spaces and matrices. This method uses tilting theory and the explicit knowledge of indecomposable modules over the corresponding canonical algebra of domestic type. Further, if K is algebraically closed we obtain all indecomposable representations for Γ. For the case that Γ is of type e Dn, n ≥ 4, with a fixed orientation, we determine all indecomposable preprojective representations. Moreover, in the case e E6 we present the most complicated indecomposable preprojective representations of rank 3.