On the nilpotency of certain subalgebras of Kac-Moody algebras of type A_N^{(r)}

Abstract

Let ${\frak g}={\frak g} (A) ={\frak N}_{-} \,\oplus \,{\frak h} \,\oplus \,\frak N_{+}$ be a symmetrizable Kac-Moody algebra with the indecomposable generalized Cartan matrix $A$ and $W$ be its Weyl group. Let $\theta$ be the highest root of the corresponding finite dimensional simple Lie algebra $\gg$ of ${\frak g}$. For the type $A_N^{(r)}$, we give an element $w_0 \in W$ such that $w_0^{-1}(\DD_+) = \DD_-$. And then we prove that the degree of nilpotency of the subalgebra $S_{w} = {\frak N}_{+} \,\cap \,w \left({\frak N}_{-}\right)$ is greater than or equal to ht$\theta + 1$.