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Journal of Lie Theory 18 (2008), No. 4, 897--914
Copyright Heldermann Verlag 2008

Lie Algebras of Hamiltonian Vector Fields and Symplectic Manifolds

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


\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 $\R^{2n}$.

Keywords: Leibniz homology, symplectic manifolds, symplectic invariants.

MSC: 17B56, 53D05, 17A32

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