On the Riemann-Lie algebras and Riemann-Poisson Lie groups

2003/10/18 by Mohamed Boucetta, Boucetta, Mohamed
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Symplectic Geometry (math.SG) #math.DG #math.SG

paper · pdf · doi:10.48550/arxiv.math/0310293

17 pages

arxiv created 2003/10/18 · arxiv updated 2009/12/01

Abstract

A Riemann-Lie algebra is a Lie algebra \cal G such that its dual \cal G^* carries a Riemannian metric compatible (in the sense introduced by th author in C. R. Acad. Paris, t. 333, Série I, (2001) 763-768) with the canonical linear Poisson sructure of \cal G^*. The notion of Riemann-Lie algebra has its origins in the study, by the author, of Riemann-Poisson manifolds (see Preprint math.DG/0206102 to appear in Differential Geometry and its Applications). In this paper, we show that, for a Lie group G, its Lie algebra \cal G carries a structure of Riemann-Lie algebra iff G carries a flat left-invariant Riemannian metric. We use this characterization to construct a huge number of Riemann-Poisson Lie groups (a Riemann-Poisson Lie group is a Poisson Lie group endowed with a left-invariant Riemannian metric compatible with the Poisson structure).

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