# Talk:Poincaré–Birkhoff–Witt theorem

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• The point there is a unique reduction is missing;
• The fact that there is a reduction at all is not at all clear by applying the structure equations. One needs something like induction on the degree of the canonical monomial; thus
${\displaystyle AuvB=AvuB+\sum _{z}c_{uvz}AzB\quad }$

where the suummands

${\displaystyle \sum _{z}c_{uvz}AzB\quad }$

have lower degree and thus are reducible by the induction hypothesis.

Including this on the other hand seems overkill.CSTAR 23:14, 6 Sep 2004 (UTC)

## History

I am told that neither Henri Poincaré nor Birkhoff nor Witt gave a correct proof. I say the article should say a few words about the history of the theorem and who eventually proved it. At the very least the article should have links to the mathematicians above! Mhym 04:54, 22 March 2006 (UTC)

It would be nice mention in the article who the theorem is named for, with links to their own articles.--BillFlis 16:55, 31 March 2006 (UTC)
Not only did Poincare prove this theorem in 1900, he actually proved much stronger theorem. I will try to clean up this article. DR2006kl (talk) 13:30, 26 January 2009 (UTC)
I am far from convinced that Poincaré's proof as given in http://smf.emath.fr/Publications/RevueHistoireMath/5/pdf/smf_rhm_5_249-284.pdf is correct. On page 22 of the arXiv version ( http://arxiv.org/abs/math/9908139 ) they write:
"The first four chains are of the form ${\displaystyle U_{1}=XH_{1},\ U'_{1}=H'_{1}Z,\ U_{2}=YH_{2},\ U'_{2}=H'_{2}T}$, where each chain ${\displaystyle H_{1},\ H'_{1},\ H_{2},\ H'_{2}}$ is a closed chain of degree p - 1; therefore by induction, each is the head of an identically zero regular sum. It follows that ${\displaystyle U_{1},\ U'_{1},\ U_{2},\ U'_{2}}$ are identically zero, and therefore each of them can be considered as the head of an identically zero regular sum of degree p."
I don't understand the "It follows that ${\displaystyle U_{1},\ U'_{1},\ U_{2},\ U'_{2}}$ are identically zero" part. This seems to be equivalent to ${\displaystyle H_{1}=H'_{1}=H_{2}=H'_{2}=0}$, which I don't believe (the head of an identically zero regular sum isn't necessarily zero). Then again, the authors are only using the weaker assertion that each of ${\displaystyle U_{1},\ U'_{1},\ U_{2},\ U'_{2}}$ is the head of an identically zero regular sum of degree p, and maybe they HAVE some good argument for that which I just don't see. -- Darij (talk) 09:02, 26 May 2011 (UTC)

## Birkhoff: father/son

I have a clear impression that actually the wiki-ref. goes to the wrong Birkhoff: there are two of them, father (George David) and son (Garret), and the work cited in the references is the one of G.D. Birkhoff. I'll correct the link. --Burivykh (talk) 08:29, 16 July 2010 (UTC)

I have just corrected the attribution. The 1937 paper is by the son, Garrett Birkhoff. See the original paper on JSTOR (http://www.jstor.org/stable/1968569). See also a history of this theorem on ScienceDirect (http://dx.doi.org/10.1016/S0723-0869(04)80010-0). -- J.P. Martin-Flatin (talk) 14:23, 31 August 2014 (UTC)

## Flatness condition

quote: More generally, the PBW theorem as formulated above extends to cases such as where (1) L is a flat K-module, (2) L is torsion-free as an abelian group, (3) L is a direct sum of cyclic modules (or all its localizations at prime ideals of K have this property), or (4) K is a Dedekind domain. See, for example, the 1969 paper by Higgins for these statements.

I don't think condition (1) appears ever in Higgins. It does with "projective" instead of "flat", which is weaker (projectives are always flat, but not conversely). Flatness emerges only at the end of the paper, with respect not to modules but to ring extensions. Am I missing something? -- Darij (talk) 22:01, 19 April 2011 (UTC)

Yes, I was missing something. By Lazard's theorem, a flat module is always the direct limit of some free modules. And by Theorem 8 in Higgins's paper, this yields that the PBW theorem holds whenever L is a free K-module. -- Darij (talk) 09:02, 26 May 2011 (UTC)