Stable representations of quivers

Let Q be a finite quiver without oriented cycles and let kQ the path algebra of Q over an algebraically closed field k. We investigate stable finite dimensional representations of Q. That is for a fixed dimension vector d and a fixed weight θ we consider θ-stable representations of Q with dimension vector d. If we wish to compare also representations with different dimension vectors, then it is more convenient to consider a slope μ instead of a weight θ. In particular, we apply the results of Harder–Narasimhan on natural filtrations associated to any fixed slope μ to the category of representations of Q. Further we introduce the wall system for weights with respect to a fixed dimension vector d and consider several examples.