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On Lie groups with left-invariant symplectic or Kählerian structures

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Abstract

Left-invariant symplectic structure on a group G; properties of the corresponding Lie algebra g. A unimodular symplectic Lie algebra has to be solvable (see [1]). Symplectic subgroups and left-invariant Poisson structures on a group. Affine Poisson structures: an affine Poisson structure associated to g and admitting g * as a unique leaf corresponds to a unimodular symplectic Lie algebra and the associate group is right-affine. If G is unimodular and endowed with a left-invariant metric g, harmonic theory for the left-invariant forms. Kählerian group is metabelian and Riemannianly flat. Decomposition of a simply connected Kählerian group. A symplectic group admitting a left-invariant metric with a nonnegative Ricci curvature is unimodular and admits a left-invariant flat Kählerian structure.

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Notes and References

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Authors and Affiliations

  1. Physique-Mathématique, Collège de France, F-75231, Paris Cedex 05, France

    André Lichnerowicz

  2. Département de Mathématiques, Université de Sciences et Techniques de Languedoc, Place Eugène Bataillon, F-34060, Montpellier Cedec, France

    Alberto Medina

Authors
  1. André Lichnerowicz
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  2. Alberto Medina
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Lichnerowicz, A., Medina, A. On Lie groups with left-invariant symplectic or Kählerian structures. Lett Math Phys 16, 225–235 (1988). https://doi.org/10.1007/BF00398959

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  • DOI: https://doi.org/10.1007/BF00398959

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