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Root polytopes and Borel subalgebras. (English) Zbl 1337.17011
Summary: Let $$\Phi$$ be a finite crystallographic irreducible root system and $${\mathcal P}_{\Phi}$$ be the convex hull of the roots in $$\Phi$$. We give a uniform explicit description of the polytope $${\mathcal P}_{\Phi}$$, 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.

##### MSC:
 17B22 Root systems 17B20 Simple, semisimple, reductive (super)algebras 52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.)
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