Compatible Kähler structures on pseudo-Hermitian quadratic Lie algebras

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Abstract
We study Lie algebras endowed with a complex structure admitting, at the same time, a pseudo-Hermitian quadratic structure and a (possibly indefinite) Kähler structure. These algebras have an invertible derivation that is skewsymmetric with respect to the quadratic structure and commutes with the complex structure. We propose several methods to construct such Lie algebras and use a generalized method of double extension to provide an inductive description of all of them. This double extension method allows us to prove that Lorentz-Hermitian quadratic Lie algebras cannot admit a Kähler structure for the same complex structure unless they are abelian. Additionally, we show that there is only one non-abelian Lie algebra of that kind for dimension less than or equal to 8 and prove that 2-step nilpotent ones with quadratic metrics of index 4 are necessarily either abelian or a trivial central extension of the aforementioned 8-dimensional Lie algebra

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