Glorious pairs of roots and Abelian ideals of a Borel subalgebra

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 $\mathfrak b$.