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 g𝔤 can attain, when its Levi subalgebra gL𝔤� is simple. We establish the structure of the nilradical of a perfect Lie algebra g𝔤, as a gL𝔤�-module, and determine the possible Lie algebra structures that one such g𝔤 admits. We prove that, as a gL𝔤�-module, the nilradical must decompose into at least four simple modules. We explicitly calculate the semisimple derivations of a perfect Lie algebra g𝔤 with Levi subalgebra gL=sl2(C)𝔤�=𝔰�2(ℂ) and give necessary conditions for g𝔤 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.

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