Flatness for the moment map for representations of quivers

We study the flatness for the moment map associated to the cotangent bundle of the space of representations of a quiver Q. If the associated moment map of a root is flat, then we call the root a flat root. We first study the flat roots in the fundamental set of a quiver Q. Then we give an explicit description of all the flat roots of Q. This description is obtained by using the natural class of (−1)-reflections, which are introduced in this paper. We also show that there are only a finite number of flat roots in each orbit under the Weyl-group of the quiver Q.