Glorious pairs of roots and Abelian ideals of a Borel subalgebra
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Glorious pairs of roots and Abelian ideals of a Borel subalgebra Dmitri I. Panyushev1 Received: 3 August 2018 / Accepted: 4 October 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract Let g be a simple Lie algebra with a Borel subalgebra b. Let + be the corresponding (po)set of positive roots and θ the highest root. A pair {η, η } ⊂ + is said to be glorious, if η, η are incomparable and η + η = θ . Using the theory of abelian ideals of b, we (1) establish a relationship of η, η to certain abelian ideals associated with long simple roots, (2) provide a natural bijection between the glorious pairs and the pairs of adjacent long simple roots (i.e., some edges of the Dynkin diagram), and (3) point out a simple transform connecting two glorious pairs corresponding to the incident edges in the Dynkin diagram. In types DE, we prove that if {η, η } corresponds to the edge through the branching node of the Dynkin diagram, then the meet η ∧ η is the unique maximal non-commutative root. There is also an analogue of this property for all other types except type A. As an application, we describe the minimal non-abelian ideals of b. Keywords Root system · Borel subalgebra · Abelian ideal · Adjacent simple roots Mathematics Subject Classification 17B20 · 17B22 · 06A07 · 20F55
Introduction Let g be a complex simple Lie algebra, with a Cartan subalgebra t and a triangular decomposition g = u ⊕ t ⊕ u− . Here b = u ⊕ t is a fixed Borel subalgebra. In this note, we present new combinatorial properties of root systems and abelian ideals of b. Our setting is always combinatorial, i.e., the abelian ideals of b, which are sums of root spaces of u, are identified with the corresponding sets of positive roots.
This research is partially supported by the R.F.B.R. Grant No. 16-01-00818.
B 1
Dmitri I. Panyushev [email protected] Institute for Information Transmission Problems of the R.A.S, Bolshoi Karetnyi per. 19, Moscow, Russia 127051
123
Journal of Algebraic Combinatorics
Let be the root system of (g, t) in the R-vector space V = t∗R , + the set of positive roots in corresponding to u, the set of simple roots in + , and θ the highest root in + . We regard + as poset with the usual partial ordering ‘’. An ideal of (+ , ) is a subset I ⊂ + such that if γ ∈ I , ν ∈ + , and ν + γ ∈ + , then ν + γ ∈ I . The set of minimal elements of I is denoted by min(I ). Write also max(+ \ I ) for the set of maximal elements of + \ I . An ideal I is abelian, if / + for all γ , γ ∈ I . γ + γ ∈ Let Ad (resp. Ab) be the set of all (resp. all abelian) ideals of + . Then Ab ⊂ Ad and we think of both sets as posets with respect to inclusion. The ideal generated by γ ∈ + is I γ = {ν ∈ + | ν γ } ∈ Ad, and γ is said to be commutative, if I γ ∈ Ab. Write + com for the set of all commutative roots. For any , this subset is explicitly described in [10, Theorem 4.4]. Note that + com ∈ Ad. A pair {η, η } ⊂ + is said to be glorious, if η, η are incomparable in the poset (+ , ), i.
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