References: Infinite dimensional Lie algebras 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).   
 A: 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. 
A: This questions seems to be closely related to the study of classical integrable systems and (mathematical) quantization. Am I right?
Maybe some literature on these subjects can be useful. 
For example:


*

*Perelomov "Integrable systems of classical mechanics and Lie algebras"

*Kirillov "Lectures on the orbit method" (ISBN   0821835300, 9780821835302)

A: Lie algebras of vector fields were treated in some 1970-1980 works of Kac and Rudakov:
Kac: 


*

*Simple irreducible graded Lie algebras of finite growth, Math. USSR Izv. 2 (1968), N6, 1271-1311 DOI:10.1070/IM1968v002n06ABEH00072 (this is the (famous) paper where the Kac-Moody algebras were introduced, but it contains also material about Lie algebras of vector fields)

*Description of filtered Lie algebras with which graded Lie algebras of Cartan type are associated, Math. USSR Izv. 8 (1974), 801-835 MR:51#5685 DOI:10.1070/IM1974v008n04ABEH002128 ZBL:0317.17002


Rudakov:


*

*Groups of automorphisms of infinite-dimensional simple Lie algebras, Math. USSR Izv. 3 (1969), 707-722 ZBL:0222.17014 DOI:10.1070/IM1969v003n04ABEH000798 

*Subalgebras and automorphisms of Lie algebras of Cartan type, Funct. Anal. Appl. 20 (1986), 72-73 ZBL:0594.17015


I think lot of information you are interested in contained there, albeit maybe not in the most explicit form.
Also, some works about finite-dimensional characteristic $p$ counterparts of these algebras (so-called  Lie algebras of Cartan type) - notably papers by Skryabin in Comm. Algebra in mid 1990s, book by Strade - start with a pretty much general context (arbitrary field, arbitrary dimension) and may be also relevant for your purpose.
A: One should certainly mention
A. Lichnerowicz, L'Algèbre de Lie des automorphisme infinitésimaux symplectiques, Symp. Math. XIV 11-24, Academic Press 1974.
A. Lichnerowicz Les variétés de Poisson et leurs algèbres de Lie associées, J. Differential Geom. 12 (1977), 253–300.
I do not have the chance to do it at present, but I remember some other refs by Lichnerowicz at around the same years. Being in French they're often forgot in the literature.
