
Journal of Lie Theory 15 (2005), No. 1, 183195 Copyright Heldermann Verlag 2005 On the RiemannLie Algebras and RiemannPoisson Lie Groups Mohamed Boucetta Faculté des Sciences et Techniques Gueliz, BP 549, Marrakech, Morocco boucetta@fstgmarrakech.ac.ma A RiemannLie algebra is a Lie algebra G such that its dual G* carries a Riemannian metric compatible (in the sense introduced recently by the author [C. R. Acad. Sci. Paris, Série I, 333 (2001) 763768] with the canonical linear Poisson structure of G*. The notion of RiemannLie algebra has its origins in the study, by the author, of RiemannPoisson manifolds [see Diff. Geometry Appl. 20 (2004) 279291]. In this paper, we show that, for a Lie group G, its Lie algebra G carries a structure of RiemannLie algebra iff G carries a flat leftinvariant Riemannian metric. We use this characterization to construct examples of RiemannPoisson Lie groups (a RiemannPoisson Lie group is a Poisson Lie group endowed with a leftinvariant Riemannian metric compatible with the Poisson structure). [ Fulltextpdf (171 KB)] for subscribers only. 