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

Abstract

Let G be a Lie group with Lie algebra $ \Cal G: = T_εG$ 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 $T^{\ast }G$ 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

13 pages, latex

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