Journal of Lie Theory

A Riemann-Lie 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) 763--768] with the canonical linear Poisson structure of G*. The notion of Riemann-Lie algebra has its origins in the study, by the author, of Riemann-Poisson manifolds [see Diff. Geometry Appl. 20 (2004) 279--291].
In this paper, we show that, for a Lie group G, its Lie algebra G carries a structure of Riemann-Lie algebra iff G carries a flat left-invariant Riemannian metric. We use this characterization to construct examples 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).