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Poisson manifolds with compatible pseudo-metric and pseudo-Riemannian Lie algebras. (English) Zbl 1061.53058

The author exploits a notion of compatibility between a Poisson structure \(\pi\) and a pseudo-Riemannian contravariant metric \(g\) on a manifold \(M\), introduced in a previous paper [C. R. Acad. Sci., Paris, Sér. I, Math. 333, No. 8, 763–768 (2001; Zbl 1009.53057)]. The compatibility condition takes the form \(\nabla\pi=0\), where \(\nabla\) is the Lie algebroid Levi-Civita connection on \(T^*M\) with respect to \(g\) and the Lie algebroid structure on \(T^*M\) induced by \(\pi\). In the paper under review the notion of pseudo-Riemannian Lie algebras is introduced: a Lie algebra \(\mathcal G\) with a pseudo-Riemannian metric is a pseudo-Riemannian Lie algebra if the metric is compatible with the canonical Poisson structure on the dual \(\mathcal G^*\). One can prove that the Lie algebra obtained by linearizing a pseudo-Riemannian Poisson manifold at a point turns out to be a pseudo-Riemannian Lie algebra. A classification of all pseudo-Riemannian Lie algebras in dimensions 2 and 3 is given. It is also proven that any symplectic leaf \(S\) of a Riemannian Poisson manifold is a Kähler manifold, that the holonomy group of such a regular \(S\) is finite, and that every Riemannian Poisson manifold is unimodular, i.e. its modular class vanishes. Finally, Poisson Lie groups with compatible pseudo-Riemannian metrics are studied.

MSC:

53D17 Poisson manifolds; Poisson groupoids and algebroids
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
17B99 Lie algebras and Lie superalgebras
53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)

Citations:

Zbl 1009.53057
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References:

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