Abelian ideals of a Borel subalgebra and root systems

TL;DR

This paper explores the structure of abelian ideals in Borel subalgebras of simple Lie algebras, connecting their partition to root systems, centralisers, and semilattice properties, thereby deepening understanding of their algebraic and combinatorial features.

Contribution

It relates a partition of abelian ideals to root system parametrizations, centralisers, and proves the positive roots form a join-semilattice.

Findings

Partition of abelian ideals aligns with root system properties

Poset of positive roots forms a join-semilattice

Connections established with Kostant-Peterson parameterization

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 $Ab_\mu$, 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.