Free Poisson fields and their automorphisms

@article{MakarLimanov2012FreePF,
  title={Free Poisson fields and their automorphisms},
  author={L. Makar-Limanov and U. Umirbaev},
  journal={Journal of Algebra and Its Applications},
  year={2012},
  volume={15},
  pages={1650196}
}
Let k be an arbitrary field of characteristic 0. We prove that the group of automorphisms of a free Poisson field P(x,y) in two variables x,y over k is isomorphic to the Cremona group Cr2(k). We also prove that the universal enveloping algebra P(x1,…,xn)e of a free Poisson field P(x1,…,xn) is a free ideal ring and give a characterization of the Poisson dependence of two elements of P(x1,…,xn) via universal derivatives. 
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References

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Automorphisms and derivations of free Poisson algebras in two variables
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The Freiheitssatz for Poisson algebras
Subalgebras of Free Lie Algebras
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