Abelian ideals of a Borel subalgebra and long positive roots

Abstract

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.