Representations as elements in affine composition algebras

Let be the path algebra of a Euclidean quiver over a finite field . The aim of this paper is to classify the modules with the property , where is Ringel's composition algebra. Namely, the main result says that if , then if and only if the regular direct summand of is a direct sum of modules from non-homogeneous tubes with quasi-dimension vectors non-sincere. The main methods are representation theory of affine quivers, the structure of triangular decompositions of tame composition algebras, and the invariant subspaces of skew derivations. As an application, we see that if and only if the quiver of is of Dynkin type.