Poisson manifolds with compatible pseudo-metric and pseudo-Riemannian Lie algebras
By M. Boucetta. 2002
47 citations recorded by the source corpus.
Abstract In a previous paper (C. R. Acad. Sci. Paris Ser. I 333 (2001) 763–768), the author introduced a notion of compatibility between a Poisson structure and a pseudo-Riemannian metric. In this paper, we introduce a new class of Lie algebras called pseudo-Riemannian Lie algebras. The two notions are closely related: we prove that the dual of a Lie algebra endowed with its canonical linear Poisson structure carries a compatible pseudo-Riemannian metric if and only if the Lie algebra is a pseudo-Riemannian Lie algebra. Moreover, the Lie algebra obtained by linearizing at a point a Poisson manifold with a compatible pseudo-Riemannian metric is a pseudo-Riemannian Lie algebra. We also give some properties of the symplectic leaves of such manifolds, and we prove that every Poisson manifold with a compatible Riemannian metric is unimodular. Finally, we study Poisson Lie groups endowed with a compatible pseudo-Riemannian metric, and we give the classification of all pseudo-Riemannian Lie algebras of dimension 2 and 3.
Subfields: Applied Mathematics, Mathematical Physics, Pathology and Forensic Medicine
View Paper: https://www.semanticscholar.org/paper/e752e506eb293c2f0eb700e8e149be44c10a6a46
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Citations in the corpus
- Riemannian Geometry of Lie Algebroids (2008, methodology): “Lie algebroid is a natural extension of the classical notion of Riemannian metric on a manifold and this notion appeared already in the context of Lie algebroids associated to Poisson structures (see [2, 3, 12, 13]). In this paper, we present the basic concepts related to a Riemannian structure on a”