Journal of Lie Theory

no. 4
Lie Algebras of Hamiltonian Vector Fields and Symplectic Manifolds
Journal of Lie Theory, Volume 18 (2008) no. 4, pp. 897-914
\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 $\R^{2n}$.
DOI: 10.5802/jolt.531
Classification: 17B56, 53D05, 17A32
Keywords: Leibniz homology, symplectic manifolds, symplectic invariants 
@article{JOLT_2008_18_4_a8,
     author = {J. M. Lodder},
     title = {Lie {Algebras} of {Hamiltonian} {Vector} {Fields} and {Symplectic} {Manifolds}},
     journal = {Journal of Lie Theory},
     pages = {897--914},
     year = {2008},
     volume = {18},
     number = {4},
     doi = {10.5802/jolt.531},
     zbl = {1171.17006},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.531/}
}
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J. M. Lodder. Lie Algebras of Hamiltonian Vector Fields and Symplectic Manifolds. Journal of Lie Theory, Volume 18 (2008) no. 4, pp. 897-914. doi: 10.5802/jolt.531

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