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Mar 30, 2013 at 22:09 comment added Reimundo Heluani The associated graded is a Poisson algebra and the Poisson bracket is graded there. If $\sigma(X)$ denotes the symbol of $X$, then I suppose your question is when ${\sigma(X), \sigma(Y)}\neq 0$. I'm not sure there's more you can say.
Mar 29, 2013 at 21:34 history edited Chuck Hague CC BY-SA 3.0
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Mar 29, 2013 at 21:31 comment added Chuck Hague Oh yikes, yes. That's a good point. I obviously haven't thought this through well enough.
Mar 29, 2013 at 21:23 comment added Bruce Westbury Isn't $X=Y$ a counterexample to both your proposals?
Mar 29, 2013 at 21:07 history asked Chuck Hague CC BY-SA 3.0