On the determination of 2-step solvable Lie algebra from its weight graph

Abstract

By using the concept of weight graph associated to certain nilpotent Lie algebras $\mathfrak{g}$, we find necessary and sufficient conditions for a semidirect product $\mathfrak{g}\oplus T_i$, where $T_i<T$ is a subalgebra of a maximal torus of derivations $T$ of $\mathfrak{g}$ which induces a decomposition of $\mathfrak{g}$ into one dimensional weight spaces, to be 2-step solvable. In particular we show that the semidirect product of such a Lie algebra with its torus of derivations cannot be itself 2-step solvable.

Corrected, minor typographical changes