Root Polytopes and Borel Subalgebras

Let Φ be a finite crystallographic irreducible root system and |PΦ| be the convex hull of the roots in Φ. We give a uniform explicit description of the polytope |PΦ|⁠, analyze the algebraic-combinatorial structure of its faces, and provide connections with the Borel subalgebra of the associated Lie algebra. We also give several enumerative results.