Re: Lie algebras and failure of PBW

[Johannes Huebschmann <huebschm@math.univ-lille1.fr>]

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 there is a hint at an unpublished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