
Journal of Lie Theory 16 (2006), No. 2, 371391 Copyright Heldermann Verlag 2006 Invariant PseudoKähler Metrics in Dimension Four Gabriela P. Ovando Fa.M.A.F., U. N. de Córdoba, Ciudad Universitaria, 5000 Córdoba, Argentina ovando@mate.uncor.edu 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 pseudoRiemannian metric g, for which J is a parallel. It is proved that most of these complex homogeneous spaces admit a compatible pseudoKähler Einstein metric. Ricci flat and flat metrics are determined. In particular Ricci flat unimodular pseudoKähler Lie groups are flat in dimension four. Other algebraic and geometric features are treated. A general construction of Ricci flat pseudoKähler structures in higher dimension on some affine Lie algebras is given. Walker and hypersymplectic metrics are compared. Keywords: PseudoKaehler metrics, Kaehler Lie algebras, invariant metrics, four dimensional Lie algebras. MSC: 32Q15, 32Q20, 53C55, 32M10, 57S25, 22E25 [ Fulltextpdf (228 KB)] for subscribers only. 