Root polytopes and Abelian ideals

We study the root polytope $\mathcal{P}_{\varPhi}$ of a finite irreducible crystallographic root system Φ using its relation with the Abelian ideals of a Borel subalgebra of a simple Lie algebra with root system Φ. We determine the hyperplane arrangement corresponding to the faces of codimension 2 of $\mathcal{P}_{\varPhi}$ and analyze its relation with the facets of $\mathcal{P}_{\varPhi}$ . For Φ of type A _n or C _n , we show that the orbits of some special subsets of Abelian ideals under the action of the Weyl group parametrize a triangulation of  $\mathcal{P}_{\varPhi}$ . We show that this triangulation restricts to a triangulation of the positive root polytope  $\mathcal{P}_{\varPhi}^{+}$ .