Abelian ideals of a Borel subalgebra and root systems

Abstract

ABSTRACT. Let g be a simple Lie algebra and Ab o the poset of non-trivial abelian ideals of a fixed Borel subalgebra of g. In [9], we constructed a partition Ab o = ⊔ µ Ab µ parameterised by the long positive roots of g and studied the subposets Ab µ . 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 is a join-semilattice.

Minimal inversion complete sets and maximal abelian ideals

“…Writeα for the unique simple root such that (θ,α) = 0. The corresponding maximal abelian idealâ := aα has the property that I(â) ⊂ H := {γ ∈ ∆ + | (γ, θ) > 0} [10]. Explicit computations for D n and E n (n 6) suggest that it might be true that if θ is fundamental, then Ess(F a ) ⊂ I(a) ∪ H.…”

“…Consequently, in place of the dimension of a ⊂ u + , we deal with the cardinality of I(a) ⊂ ∆ + , etc. It is proved in [10] that there is a one-to-one correspondence between the maximal abelian ideals and the long simple roots in ∆ + . As our subsequent results on MICS heavily rely on that correspondence, we recall the necessary setup.…”

“…Proofs in this article heavily rely on some further results obtained in [10,11]. For instance, the construction of F a (in the ADE case!)…”

“…For any µ ∈ ∆ + l , W contains a unique element of minimal length taking θ to µ [10, Theorem 4.1]. This element is denoted by w µ in [10] and here we write w θ,µ for it. Its inverse has the following description:…”

“…In general, the maximal abelian ideals are naturally parameterised by the long simple roots [10,Corollary 3.8]. (We say more on this correspondence in Section 1.)…”