Ignacio Bajo

Almost abelian Lie groups are those admitting a normal subgroup which is commutative and of codimension one. We determine all Lie algebras of almost abelian Lie groups admitting a left-invariant Lorentzian metric of vanishing curvature. For each of those algebras, we completely classify the set of inequivalent flat Lorentzian products.

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

We show that there are no symmetric non-zero biderivations on perfect Lie algebras of finite dimension over a field of characteristic zero. We show that this is equivalent to show that every symmetric biderivation on a finite-dimensional perfect Lie algebra over such a field with values in a finite-dimensional module vanishes identically. This answers an open question posed by M. Brešar and K. Zhao.

We give a complete classification of flat Lorentzian nilpotent Lie algebras, this is to say of pseudo‐Euclidean Lie algebras associated to nilpotent Lie groups endowed with a left‐invariant Lorentzian metric of vanishing curvature. We prove that every such a Lie algebra is a direct sum of an indecomposable flat Lorentzian Lie algebra and an abelian Euclidean summand and show that, if denotes the ‐dimensional Heisenberg Lie algebra, then the only non‐abelian Lie algebras admitting flat Lorentzian metrics which are indecomposable are and the semidirect products and , defined by some particular derivations . In all those cases we also find the equivalence classes of flat Lorentzian products.

We study nilpotent Lie algebras endowed with a complex structure and a quadratic structure which is pseudo-Hermitian for the given complex structure. We propose several methods to construct such Lie algebras and describe a method of double extension by planes to get an inductive description of all of them. As an application, we give a complete classification of nilpotent quadratic Lie algebras where the metric is Lorentz-Hermitian and we fully classify all nilpotent pseudo-Hermitian quadratic Lie algebras up to dimension 8 and their inequivalent pseudo-Hermitian metrics.

We study nilpotent Lie algebras endowed with a complex structure and a quadratic structure which is pseudo-Hermitian for the given complex structure. We propose several methods to construct such Lie algebras and describe a method of double extension by planes to get an inductive description of all of them. As an application, we give a complete classification of nilpotent quadratic Lie algebras where the metric is Lorentz-Hermitian and we fully classify all nilpotent pseudo-Hermitian quadratic Lie algebras up to dimension 8 and their inequivalent pseudo-Hermitian metrics.