Let be a simple Lie algebra and the poset of non-trivial abelian ideals of a fixed Borel subalgebra of . In , we constructed a partition parameterised by the long positive roots of and studied the subposets . 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 is a join-semilattice.
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Dmitri I. Panyushev, Abelian ideals of a Borel subalgebra and root systems. J. Eur. Math. Soc. 16 (2014), no. 12, pp. 2693–2708DOI 10.4171/JEMS/496