JournalsjemsVol. 16, No. 12pp. 2693–2708

Abelian ideals of a Borel subalgebra and root systems

  • Dmitri I. Panyushev

    Independent University of Moscow, Russian Federation
Abelian ideals of a Borel subalgebra and root systems cover
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Abstract

Let g\mathfrak g be a simple Lie algebra and Abo\mathfrak {Ab}^o the poset of non-trivial abelian ideals of a fixed Borel subalgebra of g\mathfrak g. In [8], we constructed a partition Abo=μAbμ\mathfrak {Ab}^o =\sqcup_\mu \mathfrak {Ab}_\mu parameterised by the long positive roots of g\mathfrak g and studied the subposets Abμ\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 g\mathfrak g is a join-semilattice.

Cite this article

Dmitri I. Panyushev, Abelian ideals of a Borel subalgebra and root systems. J. Eur. Math. Soc. 16 (2014), no. 12, pp. 2693–2708

DOI 10.4171/JEMS/496