Classical Yang-Baxter Equation and Left Invariant Affine Geometry on Lie Groups

Diatta, Andre ; Medina, Alberto

HAL, hal-00942337 / Harvested from HAL

Résumé

Let G be a Lie group with Lie algebra $ \Cal G: = T_\epsilon G$ 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 $r\in \wedge^2 \Cal G$ 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 \oplus \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 $\pi := r^+ - r^-$ is polynomial of degree at most 2 and the double Lie groups of $(G,\pi)$ 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

Détail

Publié le : 2002-03-19
Classification:  Classical Yang-Baxter Equation,  Left Invariant Affine Geometry,  affine connection,  Poisson Lie group,  invariant symplectic structure,  Drinfel'd Double,  coadjoint action,  Lagrangian graph,  symplecti Lie algebra,  Equation de Yang Baxter,  connexion affine invariante,  groupe de Lie,  algèbre de Lie,  structure symplectique invariante,  linéarisation de structure de Poisson,  structure hamiltonienne,  53D17; 53A15; 17B62,  [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG]