Lattices in some symplectic or affine nilpotent Lie groups

The main aim of this paper is the description of a large class of lattices in some nilpotent Lie groups, sometimes filiforms, carrying a flat left invariant linear connection and often a left invariant symplectic form. As a consequence we obtain an infinity of, non homeomorphic, compact affine or symplectic nilmanifolds. 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. (C) 2014 Published by Elsevier B.V.