  Let $G$ be a connected Lie group and $\mathfrak{g}$ its Lie algebra. We
denote by $\nabla^0$ the torsion free bi-invariant linear connection on $G$
given by $\nabla^0_XY=\frac12[X,Y],$ for any left invariant vector fields
$X,Y$. A Poisson structure on $\mathfrak{g}$ is a commutative and associative
product on $\mathfrak{g}$ for which $\mathrm{ad}_u$ is a derivation, for any
$u\in\mathfrak{g}$.
  A torsion free bi-invariant linear connections on $G$ which have the same
curvature as $\nabla^0$ is called special. We show that there is a bijection
between the space of special connections on $G$ and the space of Poisson
structures on $\mathfrak{g}$. 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 $\nabla^0$ 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.
