A Characteristic Map for Symplectic Manifolds

TL;DR

This paper introduces a local characteristic map linking cohomology of Hamiltonian vector fields to the topology of symplectic manifolds, revealing algebraic structures related to the symplectic form.

Contribution

It constructs a novel local characteristic map for symplectic manifolds using Leibniz cohomology of Hamiltonian vector fields, extending algebraic understanding.

Findings

Leibniz cohomology of g_n forms an exterior algebra on the symplectic form.

The characteristic map relates local cohomology to global symplectic topology.

Leibniz homology maps connect Hamiltonian vector fields on R^{2n} to those on M.

Abstract

We construct a local characteristic map to a symplectic manifold M via certain cohomology groups of Hamiltonian vector fields. For each p in M, the Leibniz cohomology of the Hamiltonian vector fields on R^{2n} maps to the Leibniz cohomology of all Hamiltonian vector fields on M. For a particular extension g_n of the symplectic Lie algebra, the Leibniz cohomology of g_n is shown to be an exterior algebra on the canonical symplectic two-form. The Leibniz homology of g_n then maps to the Leibniz homology of Hamiltonian vector fields on R^{2n}.