Lie Algebras of Hamiltonian Vector Fields and Symplectic Manifolds

Jerry M. Lodder¹

¹Dept. of Mathematical Sciences, New Mexico State University, Box 30001, Las Cruces, NM 88003, U.S.A.

Journal of Lie Theory, Tome 18 (2008) no. 4, pp. 897-914

Résumé

\def\g{{\frak g}} \def\R{{\Bbb R}} 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 cohomology of this extension is then a direct summand of the Leibniz cohomology of all Hamiltonian vector fields on $\R^{2n}$.

Détail

Classification : 17B56, 53D05, 17A32
Mots-clés : Leibniz homology, symplectic manifolds, symplectic invariants