Poisson manifolds with compatible pseudo-metric and pseudo-Riemannian Lie algebras

Details Poisson manifolds with compatible pseudo-metric and pseudo-Riemannian Lie algebras Poisson manifolds with compatible pseudo-metric and pseudo-Riemannian Lie algebras

2002-06-10

arxiv.org/abs/math/0206102v1

Mathematics

Differential Geometry

13 pages

Scientific paper

The notion of Poisson manifold with compatible pseudo-metric was introduced by the author in [1]. In this paper, we introduce a new class of Lie algebras which we call a pseudo-Rieamannian Lie algebras. The two notions are strongly related: we prove that a linear Poisson structure on the dual of a Lie algebra has a compatible pseudo-metric if and only if the Lie algebra is a pseudo-Riemannian Lie algebra, and that the Lie algebra obtained by linearizing at a point a Poisson manifold with compatible pseudo-metric is a pseudo-Riemannian Lie algebra. Furthermore, we give some properties of the symplectic leaves of such manifolds, and we prove that every Poisson manifold with compatible metric (every Riemann-Lie algebra) is unimodular. As a final, we classify all pseudo-Riemannian Lie algebras of dimension 2 and 3.

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