Quasi-Poisson Manifolds
  A quasi-Poisson manifold is a G-manifold equipped with an invariant bivector field whose Schouten bracket is the trivector field generated by the invariant element in $\wedge^3 \g$ associated to an invariant inner product. We introduce the concept of the fusion for such manifolds, and we relate quasi-Poisson manifolds to the previously introduced quasi-Hamiltonian manifolds with group-valued moment maps. 
Anton Alekseev, Yvette Kosmann-Schwarzbach, Eckhard Meinrenken