Root polytopes and abelian ideals

We study the root polytope ${\mathcal{P}}_{\mathrm{\Phi }}$ of a finite irreducible crystallographic root system $\mathrm{\Phi }$ using its relation with the abelian ideals of a Borel subalgebra of a simple Lie algebra with root system $\mathrm{\Phi }$. We determine the hyperplane arrangement corresponding to the faces of codimension 2 of ${\mathcal{P}}_{\mathrm{\Phi }}$ and analyze its relation with the facets of $\mathcal P_\Phi$\mathrm{.}For$\Phi$oftype$A_n$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 ${\mathcal{P}}_{\mathrm{\Phi }}$. We show that this triangulation restricts to a triangulation of the positive root polytope ${\mathcal{P}}_{\mathrm{\Phi }}^{+}$.