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 $\{\eta ,{\eta }^{\prime }\}\subset {\Delta }^{+}$ is said to be glorious, if $\eta ,{\eta }^{\prime }$ are incomparable and $\eta +{\eta }^{\prime }=\theta$. Using the theory of abelian ideals of $\mathfrak{b}$, we (1) establish a relationship of $\eta ,{\eta }^{\prime }$ 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 $\mathbf{DE}$, we prove that if $\{\eta ,{\eta }^{\prime }\}$ corresponds to the edge through the branching node of the Dynkin diagram, then the meet $\eta \wedge {\eta }^{\prime }$ is the unique maximal non-commutative root. There is also an analogue of this property for all other types except type $\mathbf{A}$. As an application, we describe the minimal non-abelian ideals of $\mathfrak{b}$.

arXiv comment20 pages