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 $I⟨\succcurlyeq \gamma ⟩$. It is known that $I(\gamma {)}_{max}\subset I⟨\succcurlyeq \gamma ⟩$, 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. ${\mathcal{A}}_{\mathsf{pr}}$) is the set of simple (resp. primitive) roots, then there is a natural bijection $\Pi ⟷{\mathcal{A}}_{\mathsf{pr}}$. We also describe the set $\mathcal{A}\cap \mathcal{H}$, where $\mathcal{H}$ is the Heisenberg subset of ${\Delta }^{+}$.

9 pages, to appear in International J. of Mathematics