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there is a hint at an unpublished manuscript of R. Lyndon [1955] containing an example of a Lie algebra over an integral domain for which the statement of the PBW theorem is not true.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