We provide an explicit bijection between the ad-nilpotent ideals of a Borel subalgebra of a simple Lie algebra g and the orbits ofQ/(h + 1)Q under the Weyl group (Q being the coroot lattice and h the Coxeter number of g). From this result we deduce in a uniform way a counting formula for the ad-nilpotent ideals.1991 Mathematics Subject Classification. Primary: 17B20; Secondary: 20F55.
Abstract. Let Φ be a finite crystallographic irreducible root system and P Φ be the convex hull of the roots in Φ. We give a uniform explicit description of the polytope P Φ , 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.
This paper is devoted to a detailed study of certain remarkable posets which form a natural partition of all abelian ideals of a Borel subalgebra. Our main result is a nice uniform formula for the dimension of maximal ideals in these posets. We also obtain results on ad-nilpotent ideals which complete the analysis started in (J. Algebra 225 (2000) 130, 258 (2002) 112). (C) 2003 Elsevier Inc. All rights reserved
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