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Can we apply the theorem of quantization of Lie bialgebras of Etingof-Kazdhan to a Poisson manifold $\textbf{R}^{n}?$ Does it give something in comparison to Kontsevitch celebrated formula about star products on $\textbf{R}^{n}$.probably i am missing something very obvious, sorry for that, i am a beginner in this field

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You are much more likely to get a helpful response to your question if you include some additional information. What is the statement of the theorem that you are referring to? Do you understand the hypotheses? Are the hypotheses satisfied for the example you are referring to? The more specific you can be, the better. – MTS Apr 12 '12 at 20:42
What Poisson structure do you have in mind for R^n? In any case, it is known that Kontsevich's formula when applied to the dual to a Lie algebra does reproduce the universal enveloping algebra, but not in the BCH coordinates — you have to conjugate by the Duflo map. It is not expected that Kontsevich's quantization preserves group structures (perhaps it is known not to — I'm not sure). Note that both Etingof–Kazhdan and Kontsevich formulas are "local/formal" things: they really are about coordinate patches, and so every manifold "is" R^n. Both EK and K do have globalization results too. – Theo Johnson-Freyd Apr 12 '12 at 21:58

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