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This seems to be a question whose answer depends on whom you ask. Maybe we can come up with a final answer?

It is known that Poincaré, at least, invented something that can be called Poincaré-Birkhoff-Witt theorem (PBW theorem) in 1900. Okay, it was a version of the PBW theorem that required a basis of the Lie algebra, and it was formulated over $\mathbb R$ or $\mathbb C$ only, but this is not unexpected of a 1900 discovery, and generalizing the statement is pretty straightforward from a modern perspective (generalizing the proofs, not so much).

What is unclear is whether Poincaré gave a correct proof. Poincaré's proof appears in two places: his paper

Henri Poincaré, Sur les groupes continus, Transactions of the Cambridge Philosophical Society, 18 (1900), pp. 220–255 = Œuvres de Henri Poincaré, vol., III, Paris: Gauthier-Villars (1934), pp. 173–212,

and the modern paper

T. Ton-That, T.-D. Tran, Poincaré's proof of the so-called Birkhoff-Witt theorem, Rev. Histoire Math., 5 (1999), pp. 249-284, also arXiv:math/9908139.

I have no access to the former source, so all my knowledge of the proof comes from the latter.

Ton-That and Tran claim that Poincaré's proof was long misunderstood as wrong, while in truth it is a correct, if somewhat incomplete proof. The incompleteness manifests itself in the fact that a property of what Poincaré called "symmetric polynomials" (and what we nowadays call "symmetric tensors") was used but not proven. However, in my opinion this is not a flaw: This property (which appears as Theorem 3.3 in the paper by Ton-That and Tran and is proven there in an overly complicated, yet nice way) is simply the fact that the $k$-th symmetric power of a vector space $V$ over a field of characteristic $0$ is generated by $k$-th (symmetric) powers of elements of $V$. This fact is known nowadays and was known in 1900 (I think it lies at the heart of umbral calculus).

What I am not sure about is the actual proof of PBW. Since I don't have the original Poincaré source (nor, probably, the understanding of French required to read it), I am again drawing conclusions from the Ton-That and Tran paper. My troubles lie within this paragraph on pages 277-278:

"The first four chains are of the form

$U_1 = XH_1,\ U'_1 = H'_1Z,\ U_2 = YH_2,\ U'_2 = H'_2T$,

where each chain $H_1$, $H'_1$, $H_2$, $H'_2$ is a closed chain of degree $p - 1$; therefore by induction, each is the head of an identically zero regular sum. It follows that $U_1$, $U'_1$, $U_2$, $U'_2$ are identically zero, and therefore each of them can be considered as the head of an identically zero regular sum of degree $p$."

Question: Why does "It follow[] that $U_1$, $U'_1$, $U_2$, $U'_2$ are identically zero"? This seems to be equivalent to $H_1 = H'_1 = H_2 = H'_2 = 0$, which I don't believe (the head of an identically zero regular sum isn't necessarily zero). The authors, though, are only using the weaker assertion that each of $U_1, U'_1, U_2, U'_2$ is the head of an identically zero regular sum of degree $p$ - but this isn't obvious to me either.

What am I missing? Is this a mistake in Poincaré 1900? Or have the authors of 1 misrepresented Poincaré's argument? Has anybody else tried to decipher Poincaré's proof?

PS. I have asked more or less asked this question some months ago, but it was hidden in another question and did not receive any answer. Mailing the authors did not help either. So my last hope is a public discussion.

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  • $\begingroup$ You mentioned that for fields of char 0, the elements of k-th symmetric power of a vector space is generated by symmetric powers of vectors. Beyond char 0, is it true for divided powers? $\endgroup$
    – Z. M
    Commented Apr 9, 2023 at 10:34
  • $\begingroup$ @Z.M: Maybe for a very restrictive definition of Lie algebras. Certainly not for abelian ones, since their universal envelopes are still their symmetric algebras. $\endgroup$
    – darij grinberg
    Commented Apr 9, 2023 at 12:42

1 Answer 1

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I don't know if Poincaré proved PBW in 1900, but Alfredo Capelli did it for $\mathfrak{g}\mathfrak{l}_n$ ten years before. Here is the link to Capelli's paper (in French).

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  • $\begingroup$ Thanks, this is interesting. Is this the paper with the famed Capelli identities? $\endgroup$
    – darij grinberg
    Commented Sep 8, 2011 at 19:50
  • $\begingroup$ Yes it has these identities but I am not sure it is the first paper which contains them. There might be some earlier papers in Italian. $\endgroup$
    – Abdelmalek Abdesselam
    Commented Sep 8, 2011 at 19:53
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    $\begingroup$ This is a remarkable paper: it describes the universal enveloping algebra (probably, for the first time) and gives an explicit description of its center (all in the case of $\mathfrak{gl}_n$). Also, Capelli's German paper with the original Capelli identity appeared earlier. $\endgroup$
    – Victor Protsak
    Commented Sep 9, 2011 at 1:38

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