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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).

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I'm unaware of books that treat this material, but Alan D. Weinstein has many papers along these lines. If you can browse the titles using MathSciNet or other sources you will probably find some relevant ones. (Books dealing with infinite dimensional Lie algebras are relatively few, including the text by Kac and an AMS translation by Wakimoto, but the emphasis tends to be more on affine Lie algebras and mathematical physics.) –  Jim Humphreys Jan 31 '12 at 15:05
    
I saw this bias and that is why I decided to post the question here. It is true that I forgot to check the Poisson literature, thanks for remind me that, my bad. –  Romero Solha Jan 31 '12 at 16:06
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arxiv.org/abs/dg-ga/9703010 Geometric Quantization and No Go Theorems Viktor L. Ginzburg, Richard Montgomery - this is precisely about one of the questions which you ask... do not be afraid of title, actually there point of view on quantization is different from physicts that is why they get no-go where physicists "go". Quote: "This is a consequence of a ``no go'' theorem claiming that the entire Lie algebra of smooth functions on a compact symplectic manifold cannot be quantized, i.e., it has no essentially nontrivial finite-dimensional representations." –  Alexander Chervov Jan 31 '12 at 17:39
    
As far as I remember book "Identities of algebras and their representations" Yu. Razmyslov contains some result about Lie algebra of vector fields on affine manifolds, e.g. manifold can be restored from the Lie algebra - but I am not sure –  Alexander Chervov Jan 31 '12 at 19:05
    
I am not afraid at all, Alexander, this is exactly related to what I want. But I discussed this with Viktor and we do not share the same opinion (looking at this paper I can now understand part of his disbelief on geometric quantisation). For instance "We prove that for a broad class of manifolds, including symplectic homogeneous spaces (e.g., the sphere), such connection does not exist." is far from being true as a general statement, and the problem lies on the definition of geometric quantisation that is being used. –  Romero Solha Jan 31 '12 at 21:40
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4 Answers

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.

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I will look for this book, thanks. –  Romero Solha Jan 31 '12 at 16:02
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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)
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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.

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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.

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Actually that is my main problem: most of the literature I could find is in French. –  Romero Solha Apr 6 '12 at 15:58
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