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In general case the validity of an analogue of the Poincare-Birkhoff-Witt theorem for poisson algebras is open. For which class of Poisson algebras PBW theroem is valid?

Nil
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    Perhaps you will be luckier if you ask in MO, while providing some more context and references? Regarding the failure of the PBW theorem to hold for associative universal enveloping algebras of Poisson algebras, see https://arxiv.org/abs/1807.05873. – Pedro Jun 02 '22 at 11:54
  • Thank you for your comment. – Nil Jun 02 '22 at 13:09

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A good reference for this question is the recent paper

Poisson enveloping algebras and the Poincaré–Birkhoff–Witt theorem

by Thierry Lambre, Cyrille Ospel and Pol Vanhaecke.

In particualr it is shown that for an important class of singular Poisson algebras, the PBW theorem still holds.

Dietrich Burde
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