Abelian ideals of a Borel subalgebra and root systems

2012/05/27 by Panyushev, Dmitri I.
#17B20 #17B22 #20F55 #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1205.5983

Abstract

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_μ, 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.

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