Glorious pairs of roots and Abelian ideals of a Borel subalgebra

TL;DR

This paper explores the structure of glorious pairs of roots in simple Lie algebras, establishing their connection to abelian ideals and Dynkin diagram edges, and characterizing minimal non-abelian ideals.

Contribution

It introduces a novel relationship between glorious root pairs and abelian ideals, and provides a bijection with Dynkin diagram edges, enhancing understanding of Lie algebra structures.

Findings

Bijection between glorious pairs and pairs of adjacent long simple roots

Connection of glorious pairs to abelian ideals associated with long simple roots

Characterization of minimal non-abelian ideals in Borel subalgebras

Abstract

Let $\mathfrak g$ be a simple Lie algebra with a Borel subalgebra $\mathfrak b$. Let $\Delta^+$ be the corresponding (po)set of positive roots and $\theta$ the highest root. A pair $\{\eta,\eta'\}\subset \Delta^+$ is said to be glorious, if $\eta,\eta'$ are incomparable and $\eta+\eta'=\theta$. Using the theory of abelian ideals of $\mathfrak b$, we (1) establish a relationship of $\eta,\eta'$ 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 digram. In types ${\bf DE}$, we prove that if $\{\eta,\eta'\}$ corresponds to the edge through the branching node of the Dynkin diagram, then the meet $\eta\wedge\eta'$ is the unique maximal non-commutative root. There is also an analogue of this property for all other types except type ${\bf A}$. As an application, we describe the minimal non-abelian ideals of $\mathfrak b$.