Abelian ideals and amazing roots

TL;DR

This paper classifies amazing roots in simple Lie algebras, explores their structure, and establishes a bijection between primitive roots and simple roots, revealing new insights into the algebraic and combinatorial properties of these roots.

Contribution

It provides a complete classification of amazing roots, introduces the concept of primitive roots, and establishes a bijection between primitive roots and simple roots in simple Lie algebras.

Findings

Number of primitive roots equals the rank of the Lie algebra

Set of primitive roots is in bijection with simple roots

Classification of amazing roots and their intersection with the Heisenberg subset

Abstract

Let $\mathfrak g$ be a simple Lie algebra with a Borel subalgebra $\mathfrak b$. To any long positive root $\gamma$, one associates two ideals of $\mathfrak b$: the abelian ideal $I(\gamma)_{max}$ and not necessarily abelian ideal $I\langle{\succcurlyeq}\gamma\rangle$. It is known that $I(\gamma)_{max} \subset I\langle{\succcurlyeq}\gamma\rangle$, and $\gamma$ is said to be amazing if the equality holds. The set of amazing roots, $\mathcal A$, is closed under the operation `$\vee$' in $\Delta^+$, and $\gamma\in\mathcal A$ is said to be primitive, if it cannot be written as $\gamma_1\vee\gamma_2$ with incomparable amazing roots $\gamma_1,\gamma_2$. We classify the amazing roots and notice that the number of primitive roots equals $\mathsf{rk}(\mathfrak g)$. Moreover, if $\Pi$ (resp. $\mathcal A_{\sf pr}$) is the set of simple (resp. primitive) roots, then there is a natural bijection $\Pi\longleftrightarrow \mathcal A_{\sf pr}$. We also describe the set $\mathcal A\cap{\mathcal H}$, where ${\mathcal H}$ is the Heisenberg subset of $\Delta^+$.