Lattices in some Symplectic or Affine Nilpotent Lie groups

TL;DR

This paper classifies lattices in certain nilpotent Lie groups with flat invariant connections and symplectic forms, leading to numerous non-homeomorphic compact affine and symplectic manifolds, and explores their geometric properties.

Contribution

It provides a comprehensive description of lattices in specific nilpotent Lie groups with invariant structures and links these to symplectic reduction and solutions of the classical Yang-Baxter equation.

Findings

Classification of lattices in symplectic and affine nilpotent Lie groups.

Existence conditions for invariant symplectic forms on Heisenberg groups.

Connection between lattices and solutions to the classical Yang-Baxter equation.

Abstract

The main aim of this paper is the description of a large class of lattices in some nilpotent Lie groups, sometimes filiformes, carrying a flat left invariant linear connection anf often a left invariant symplectic form. As a consequence we obtain an infinity of, non homeomorphic, compact affine or symplectic manifolds. 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 suffisant condition for the existence of a left invarian symplectic form, is given. In the symplectic case we show a that a lattice in the group determines naturally lattices in the double Lie group corresponding to any solution of the classical Yang-Baxter equation.