Abelian ideals and amazing roots

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 $\Delta^+$.