Geodesic
On the Riemann-Lie Algebras and Riemann-Poisson Lie Groups
Journal of Lie Theory, Tome 15 (2005) no. 1, pp. 183-195

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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).
Classification : 53D17, 17B63
Mots-clés : Poisson Lie group, Riemann-Poisson manifold, Yang-Baxter equation 
M. Boucetta. On the Riemann-Lie Algebras and Riemann-Poisson Lie Groups. Journal of Lie Theory, Tome 15 (2005) no. 1, pp. 183-195. http://geodesic.mathdoc.fr/item/JOLT_2005_15_1_a11/
@article{JOLT_2005_15_1_a11,
     author = {M. Boucetta},
     title = {On the {Riemann-Lie} {Algebras} and {Riemann-Poisson} {Lie} {Groups}},
     journal = {Journal of Lie Theory},
     pages = {183--195},
     year = {2005},
     volume = {15},
     number = {1},
     zbl = {1077.53065},
     url = {http://geodesic.mathdoc.fr/item/JOLT_2005_15_1_a11/}
}
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