Abelian ideals of a Borel subalgebra and root systems, II

TL;DR

This paper explores the combinatorial structure of abelian ideals within Borel subalgebras of simple Lie algebras, focusing on root system partitions, unions of maximal ideals, and lattice properties.

Contribution

It advances understanding of the combinatorial and lattice-theoretic properties of abelian ideals in simple Lie algebras, extending previous work to new classes.

Findings

Union of arbitrary maximal abelian ideals described for all simple Lie algebras except sl_n

Characterization of greatest lower bounds of positive roots

Identification of modular lattice subposets within positive roots

Abstract

Let $\mathfrak g$ be a simple Lie algebra with a Borel subalgebra $\mathfrak b$ and $\mathfrak{Ab}$ the set of abelian ideals of $\mathfrak b$. Let $\Delta^+$ be the corresponding set of positive roots. We continue our study of combinatorial properties of the partition of $\mathfrak{Ab}$ parameterised by the long positive roots. In particular, the union of an arbitrary set of maximal abelian ideals is described, if $\mathfrak g\ne\mathfrak{sl}_n$. We also characterise the greatest lower bound of two positive roots, when it exists, and point out interesting subposets of $\Delta^+$ that are modular lattices.