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I am looking for any places in the literature where an author has had occasion to consider Poisson structures on infinite-dimensional algebro-geometric objects, e.g. ind-varieties or proalgebraic groups (or more specifically Kac-Moody groups). There are various analytic instances I know of where one can find a discussion of Poisson structures on infinite-dimensional manifolds modeled on topological vector spaces, but I know of no such algebraic discussions and would be interested to hear if someone knows of circumstances where such things have come up.

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Perhaps the definition of "topological coisson algebra" from Beilinson's "Remarks on topological algebras" is relevant for you. It's the definition of Poisson algebra in the setting of Tate vector spaces. –  Moosbrugger Nov 8 '11 at 3:27
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I should add: regular functions on a loop group naturally form a Tate space -- this is so for sufficiently nice ind-schemes (though ind-infinite type is allowed, and accounts for half of the semi-infinity of the Tate space). And the completed symmetric algebra of $\mathfrak{g}((t))$ forms a topological coisson algebra in Beilinson's sense. –  Moosbrugger Nov 8 '11 at 13:43

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