Re: Lie algebras and failure of PBW

[Michael Barr <barr@math.mcgill.ca>]

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 there is a hint at an unpublished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
>
>
>
>
>