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 ${\mathfrak{Ab}}^o={\bigsqcup }_{\mu }{\mathfrak{Ab}}_{\mu }$ 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.