Root polytopes and Borel subalgebras

Details Root polytopes and Borel subalgebras Root polytopes and Borel subalgebras

2012-03-04

arxiv.org/abs/1203.0756v1

Mathematics

Combinatorics

20 pages

Scientific paper

Let \Phi be a finite crystallographic irreducible root system and P be the convex hull of the roots in \Phi. We provide a uniform description of the polytope P. Assume that \Phi is the set of roots of the complex finite simple Lie algebra g with respect to the Cartan subalgebra h, and fix any Borel subalgebra b of g containing h. We find a natural bijection between the set of the orbits of the faces of P under the action of the Weyl group and a distinguished set of abelian ideals of b. By means of this bijection, we obtain a bijection between the set of the orbits of the faces and a distinguished set of irreducible subsystems of the affine root system associated with \Phi.

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