Dear Michael Thank you for your message. My message was perhaps a bit cryptic. By statement of the PBW theorem I mean that, essentially, relative to the PBW filtration of the universal algebra UL of the Lie algebra L, the canonical algebra morphism from the symmetric algebra SL to the associated graded object E^0(UL) is an isomorphism. This then implies that the canonical map from L to UL is injective. More precisely: The universal algebra UL and the symmetric algebra SL both acquire filtered cocommutative coalgebra structures, and the canonical morphism SL --> E^0(UL) is one of Hopf algebras. One way to make precise the statement of the PBW theorem is to require the existence of an isomorphism UL --> SL of filtered coalgebras such that the associated graded morphism E^0(UL) --> SL is the inverse to the canonical morphism SL --> E^0(UL). Certainly the freeness of the Lie algebra is enough to guarantee the statement of the PBW theorem. More generally, L projective as a module over the ground ring still suffices I guess. Indeed, the arguments you give in Subsection 5.3 of your 1996 JPAA algebra paper imply this. Best regards Johannes On Wed, 6 May 2009, Michael Barr wrote:
It is not entirely clear what the PBW theorem is supposed to say over an arbitrary ring. Cartan-Eilenberg prove that if g is a K-free Lie algebra (K is an arbitrary ring with 1), then the enveloping algebra is K-free and on the same sort of basis as when K is a field (assume the basis is ordered, then you can take the set of increasing sequences as the basis of g^e). Although they don't, it is simple to show that if g is K-projective, so is g^e, although the idea of a basis is no longer meaningful. If g is an arbitrary K-Lie algebra, then I have no idea what a PBW theorem could say.
Michael
On Wed, 6 May 2009, Johannes Huebschmann wrote:
Dear Friends and Colleagues
On p. 331 of
Magnus-Karras-Solitar, Combinatorial group theory
there is a hint at an unpublished manuscript of R. Lyndon [1955] containing an example of a Lie algebra over an integral domain for which the statement of the PBW theorem is not true. I did not find this example in the literature not did I find any other hint at it. Does anybody know anything about it?
Many thanks in advance
Johannes