Journal of Lie Theory

Four dimensional simply connected Lie groups admitting a pseudo Kähler metric are determined. The corresponding Lie algebras are modelled and the compatible pairs (J, ω) are parametrized up to complex isomorphism (where J is a complex structure and ω is a symplectic structure). Such structure gives rise to a pseudo-Riemannian metric g, for which J is a parallel. It is proved that most of these complex homogeneous spaces admit a compatible pseudo-Kähler Einstein metric. Ricci flat and flat metrics are determined. In particular Ricci flat unimodular pseudo-Kähler Lie groups are flat in dimension four. Other algebraic and geometric features are treated. A general construction of Ricci flat pseudo-Kähler structures in higher dimension on some affine Lie algebras is given. Walker and hypersymplectic metrics are compared.