Abelian ideals of a Borel subalgebra and root systems

Abstract

Let $\mathfrak g$ be a simple Lie algebra and $\mathfrak {Ab}^o$ the poset of non-trivial abelian ideals of a fixed Borel subalgebra of $\mathfrak g$. In [8], we constructed a partition $\mathfrak {Ab}^o =\sqcup_\mu \mathfrak {Ab}_\mu$ parameterised by the long positive roots of $\mathfrak g$ and studied the subposets $\mathfrak {Ab}_\mu$. In this note, we show that this partition is compatible with intersections, relate it to the Kostant-Peterson parameterisation and to the centralisers of abelian ideals. We also prove that the poset of positive roots of $\mathfrak g$ is a join-semilattice.