Lattices in some symplectic or affine nilpotent Lie groups

We review some new facts about the geometry of compact symplectic nilmanifolds and we describe symplectic reduction for these manifolds. For the Heisenberg-Lie group, defined over a local associative and commutative finite dimensional real algebra, a necessary and sufficient condition for the existence of a left invariant symplectic form, is given. Finally in the symplectic case we show that a lattice in the group determines naturally lattices in the double Lie group corresponding to any solution of the classical Yang-Baxter equation.