Special bi-invariant linear connections on Lie groups and finite dimensional Poisson structures

Let G be a connected Lie group and g its Lie algebra. We denote by ∇0 the torsion free bi-invariant linear connection on G given by ∇X0Y=12[X,Y], for any left invariant vector fields X,Y. A Poisson structure on g is a commutative and associative product on g for which adu is a derivation, for any u∈g. A torsion free bi-invariant linear connections on G which have the same curvature as ∇0 are called special. We show that there is a bijection between the space of special connections on G and the space of Poisson structures on g. We compute the holonomy Lie algebra of a special connection and we show that the Poisson structures associated to special connections which have the same holonomy Lie algebra as ∇0 possess interesting properties. Finally, we study Poisson structures on a Lie algebra and we give a large class of examples which gives, of course, a large class of special connections.