A Characteristic Map for Symplectic Manifolds

2008/01/22 by Jerry Lodder, Lodder, Jerry
Mathematics · #17A32 #17B56 #53D05 #Advanced Algebra and Geometry #Advanced Topics in Algebra #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT) #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.0801.3446

openalex publication_date 2008/01/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 R²ⁿ 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 R²ⁿ.

Citations

• Universal enveloping algebras of Leibniz algebras and (co)homology

Related

• The Euler-Lagrange Cohomology Groups on Symplectic Manifolds
• The Generalized Liouville's Theorems via Euler-Lagrange Cohomology Groups on Symplectic Manifold
• Contact Homology of Hamiltonian Mapping Tori
• Leibniz algebras on symplectic plane and cohomological vector fields
• Characteristic Classes in Symplectic Topology