Abelian ideals and amazing roots

2017/11/11 by Panyushev, Dmitri I.
#06A07 #17B20 #17B22 #20F55 #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1711.04129

Abstract

Let \mathfrak g be a simple Lie algebra with a Borel subalgebra \mathfrak b. To any long positive root γ, one associates two ideals of \mathfrak b: the abelian ideal I(γ)_max and not necessarily abelian ideal I⟨\succcurlyeqγ⟩. It is known that I(γ)_max ⊂ I⟨\succcurlyeqγ⟩, and γ is said to be amazing if the equality holds. The set of amazing roots, \mathcal A, is closed under the operation `\vee' in Δ⁺, and γ∈\mathcal A is said to be primitive, if it cannot be written as γ₁\veeγ₂ with incomparable amazing roots γ₁,γ₂. We classify the amazing roots and notice that the number of primitive roots equals rk(\mathfrak g). Moreover, if Π (resp. \mathcal A_{\sf pr}) is the set of simple (resp. primitive) roots, then there is a natural bijection Π\longleftrightarrow \mathcal A_{\sf pr}. We also describe the set \mathcal A∩\mathcal H, where \mathcal H is the Heisenberg subset of Δ⁺.

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