%0 Journal Article
%T Abelian ideals of a Borel subalgebra and root systems
%A Dmitri I. Panyushev
%J Mathematics
%D 2012
%I arXiv
%X Let $g$ be a simple Lie algebra and $Ab$ the poset of non-trivial abelian ideals of a fixed Borel subalgebra of $g$. In 2003 (IMRN, no.35, 1889--1913), we constructed a partition of $Ab$ into the subposets $Ab_\mu$, parameterised by the long positive roots of $g$, and established some properties of these subposets. In this note, we show that this partition is compatible with intersections, relate it to the Kostant-Peterson parameterisation of abelian ideals and to the centralisers of abelian ideals. We also prove that the poset of positive roots of $g$ is a join-semilattice.
%U http://arxiv.org/abs/1205.5983v2