Poisson enveloping algebras and the Poincaré–Birkhoff–Witt theorem

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 Poincaré–Birkhoff–Witt theorem (PBW theorem). In the case of a Poisson algebra (A,⋅,{⋅,⋅}) over a commutative ring R (with unit), Rinehart's result boils down to the statement that if A is smooth (as an algebra), then gr(U(A)) and Sym(Ω(A)) are isomorphic as graded algebras; in this formula, U(A) stands for the Poisson enveloping algebra of A and Ω(A) is the A-module of Kähler differentials of A (viewing 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 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. Throughout the paper we give several examples and present some first applications of the main theorem; applications to deformation theory and to Poisson and Hochschild (co-) homology will be worked out in a future publication.