Résumé
Manuscripta Math. 114 (2004), no. 4, 477-486 Let G be a Lie group with Lie algebra \Cal G: = T_(∊) GandT^(*)G = \Cal G^(*) ⋈ Gits cotangent bundle considered as a Lie group, where G acts on\Cal G^(*)via the coadjoint action. We show that there is a 1-1 correspondance between the skew-symmetric solutionsr∈ ∧² \Cal Gof the Classical Yang-Baxter Equation in G, and the set of connected Lie subgroups ofT^(*)Gwhich carry a left invariant affine structure and whose Lie algebras are lagrangian graphs in \Cal G ⊕ \Cal G^(*) . An invertible solution r endows G with a left invariant symplectic structure and hence a left invariant affine structure. In this case we prove that the Poisson Lie tensorπ := r⁺ - r⁻is polynomial of degree at most 2 and the double Lie groups of(G,π)also carry a canonical left invariant affine structure. In the general case of (non necessarly invertible) solutions r, we supply a necessary and suffisant condition to the geodesic completness of the associated affine structure