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

In this paper, we investigate the question of the lowest possible dimension that a sympathetic Lie algebra can attain, when its Levi subalgebra L is simple. We establish the structure of the nilradical of a perfect Lie algebra , as a L-module, and determine the possible Lie algebra structures that one such admits. We prove that, as a L-module, the nilradical must decompose into at least four simple modules. We explicitly calculate the semisimple derivations of a perfect Lie algebra with Levi subalgebra L = 2(ℂ) and give necessary conditions for 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%27s decomposition. If the nilradical has four simple modules, we show that a sympathetic Lie algebra has dimension greater or equal than 25.

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