Papers โบ On the non-existence of sympathetic Lie algebras with dimension less than 25
On the non-existence of sympathetic Lie algebras with dimension less than 25
A. L. Garcia-Pulido, G. Salgado
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In this article we investigate the question of the lowest possible dimension that a sympathetic Lie algebra ๐ค can attain, when its Levi subalgebra ๐ค_L is simple. We establish the structure of the nilradical of a perfect Lie algebra ๐ค, as a ๐ค_L-module, and determine the possible Lie algebra structures that one such ๐ค admits. We prove that, as a ๐ค_L-module, the nilradical must decompose into at least 4 simple modules. We explicitly calculate the semisimple derivations of a perfect Lie algebra ๐ค with Levi subalgebra ๐ค_L = ๐ฐ๐ฉโ(โ) and give necessary conditions for ๐ค 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 4 simple modules, we show that a sympathetic Lie algebra has dimension greater or equal than 25.
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