Poisson enveloping algebras and the Poincar\'e-Birkhoff-Witt theorem

TL;DR

This paper explores the structure of Poisson enveloping algebras, extending the Poincaré-Birkhoff-Witt theorem to certain singular Poisson algebras and providing new constructions in various contexts.

Contribution

It introduces new methods for constructing Poisson enveloping algebras and demonstrates the PBW theorem's validity for a class of singular Poisson algebras.

Findings

Several new constructions of Poisson enveloping algebras.

The PBW theorem holds for certain singular Poisson algebras.

Application to Poisson hypersurfaces of affine varieties.

Abstract

Poisson algebras are, just like Lie algebras, particular cases of Lie-Rinehart algebras. The latter were introduced by Rinehart in his seminal 1963 paper, where he also introduces the notion of an enveloping algebra and proves --- under some mild conditions --- that the enveloping algebra of a Lie-Rinehart algebra satisfies a Poincar\'e-Birkhoff-Witt theorem (PBW theorem). In the case of a Poisson algebra $({\mathcal A},\cdot,\{\cdot,\cdot\})$ over a commutative ring $R$ (with unit), Rinehart's result boils down to the statement that if $\mathcal A$ is \emph{smooth} (as an algebra), then gr$(U({\mathcal A}))$ and $\mathrm{Sym}_{\mathcal A}(\Omega({\mathcal A}))$ are isomorphic as graded algebras; in this formula, $U({\mathcal A})$ stands for the Poisson enveloping algebra of ${\mathcal A}$ and $\Omega({\mathcal A})$ is the ${\mathcal A}$-module of K\"ahler differentials of ${\mathcal A}$ (viewing ${\mathcal A}$ as an $R$-algebra). In this paper, we give several new constructions of the Poisson enveloping algebra in some general and in some particular contexts. Moreover, we show that for an important class of \emph{singular} Poisson algebras, the PBW theorem still holds. In geometrical terms, these Poisson algebras correspond to (singular) Poisson hypersurfaces of arbitrary smooth affine Poisson varieties.