2002/03/19 by Andre Diatta, Alberto Medina · 1 citation
Mathematics · #math.DG #msc:53D17 #msc:53A15 #msc:17B62
paper · pdf · doi:10.1007/s00229-004-0475-8
published as Manuscripta Math. 114 (2004), no. 4, 477-486 · 13 pages, latex
arxiv created 2002/03/19 · arxiv updated 2016/09/07
Let G be a Lie group with Lie algebra \Cal G: = TεG and T^*G = \Cal G^* \rtimes G its 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 solutions r∈ \wedge2 \Cal G of the Classical Yang-Baxter Equation in G, and the set of connected Lie subgroups of T^*G which 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 tensor π:= 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