2002/10/31 by Dmitri I. Panyushev
Mathematics · #math.RT #math.CO
published as Intern. Math.Res. Notices (2003), no.35., 1889-1913 · 20 pages, Latex2e
arxiv created 2002/11/03 · arxiv updated 2009/11/30
Let \b be a Borel subalgebra of a simple Lie algebra \g and let \Ab denote the set of all Abelian ideals of \b. We consider \Ab as poset with respect to inclusion, the zero ideal being the unique minimal element of \Ab. It was shown in my paper with G.Roehrle (Adv. Math. v.159 (2001)) that there is a one-to-one correspondence between the maximal Abelian ideals and the long simple roots of \g. But the very existence of it was demonstrated in a case-by-case fashion. Here a conceptual explanation for that empirical observation is given. The main results are: 1) there is a natural mapping τ from the set of all nontrivial Abelian ideals to the set of long positive roots; 2) If I is a maximal Abelian ideal, then τ(I) is a long simple root. Restricting τ to the set of maximal Abelian ideals yields the above-mentioned correspondence; 3) Each fibre of τ is a poset in its own right, and we prove that this fibre has a unique maximal and a unique minimal element. 4) An explicit description of the minimal and the maximal ideal corresponding to a long root is given.