What I really want are properties (if it is abelian, nilpotent, solvable, simple, or semisimple; Cartan subalgebras…) of the Lie algebra of smooth functions on a symplectic manifold $(M,\omega)$; the Lie bracket being the Poisson bracket $\{ \cdot , \cdot \}$.
The symplectic form induces a surjective Lie algebra homomorphism between $(C^\infty(M), \{ \cdot , \cdot \} )$ and the hamiltonian vector fields on $M$, which is a Lie subalgebra of the Lie algebra of vector fields. Therefore the properties of $(C^\infty(M), \{ \cdot , \cdot \} )$ are related to the Lie algebra of vector fields on $M$.
Are there any references (in English) on infinite dimensional Lie algebras treating the examples above?
Any reference dealing with a specific symplectic manifold will be very useful (specially to rule out general statements).
Kac's book on Infinite dimensional Lie algebras deals with Kac-Moody algebras, and "E. Cartan, Les groups de transformations continus, infinis, simples, C. R. Acad. Sc., t.144 (1907) 1094." is in French (I cannot read it).
Best Answer
I think the following references might be useful (copied from mathscinet)
MR0874337 (88b:17001) Fuks, D. B.(2-MOSC) Cohomology of infinite-dimensional Lie algebras. Translated from the Russian by A. B. Sosinskiĭ. Contemporary Soviet Mathematics. Consultants Bureau, New York, 1986. xii+339 pp. ISBN: 0-306-10990-5
MR1756408 Feigin, B. L.(J-KYOT-R); Fuchs, D. B.(1-CAD) Cohomologies of Lie groups and Lie algebras [MR0968446 (90k:22014)]. Lie groups and Lie algebras, II, 125–223, Encyclopaedia Math. Sci., 21, Springer, Berlin, 2000. 22E60 (17B45 17B56 22E41)
There is a chance the first reference treats the question you are interested in (I don't have the book at hand). The second one is a very readable survey of Lie group and Lie algebra cohomology.