On the non existence of sympathetic Lie algebras with dimension less than 25

Abstract

In this paper, we investigate the question of the lowest possible dimension that a sympathetic Lie algebra [Formula: see text] can attain, when its Levi subalgebra [Formula: see text] is simple. We establish the structure of the nilradical of a perfect Lie algebra [Formula: see text], as a [Formula: see text]-module, and determine the possible Lie algebra structures that one such [Formula: see text] admits. We prove that, as a [Formula: see text]-module, the nilradical must decompose into at least four simple modules. We explicitly calculate the semisimple derivations of a perfect Lie algebra [Formula: see text] with Levi subalgebra [Formula: see text] and give necessary conditions for [Formula: see text] to be a sympathetic Lie algebra in terms of these semisimple derivations. We show that there is no sympathetic Lie algebra of dimension lower than 15, independently of the nilradical’s decomposition. If the nilradical has four simple modules, we show that a sympathetic Lie algebra has dimension greater or equal than 25.