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In "my youth" I computed a finite presenation of the Poisson algebra on $S^2$ (finite presentation as a Lie algebra). In what ways might this be useful? Does this allow you to extract information that would otherwise be harder to obtain?

(1) Is it possible to compute other invariants like Lie algebra cohomology from the finite presentation ... or is it much easier to compute it directly from the full Poisson algebra.

(2) I also do not fully understand how the topology on the algebra intervenes: My presentation is actually only a presentation for a dense sub-algebra of the full Poisson algebra. In what ways can it give information about the "full" algebra. Is the full-algebra already encoded in my finite presentation?

Thank you for any suggestion and help.

Klaus

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  • $\begingroup$ You are formulating quite a few questions here, which may make people hesitate to propose answers. Aside from that, a concrete example or two would be most helpful. $\endgroup$
    – Jim Humphreys
    Mar 5, 2015 at 20:13

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