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Journal of Lie Theory 16 (2006), No. 2, 371--391
Copyright Heldermann Verlag 2006

Invariant Pseudo-Kähler Metrics in Dimension Four

Gabriela P. Ovando
Fa.M.A.F., U. N. de Córdoba, Ciudad Universitaria, 5000 Córdoba, Argentina

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.

Keywords: Pseudo-Kaehler metrics, Kaehler Lie algebras, invariant metrics, four dimensional Lie algebras.

MSC: 32Q15, 32Q20, 53C55, 32M10, 57S25, 22E25

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