So, did Poincaré prove PBW or not? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T05:43:42Z http://mathoverflow.net/feeds/question/74923 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/74923/so-did-poincare-prove-pbw-or-not So, did Poincaré prove PBW or not? darij grinberg 2011-09-08T19:12:23Z 2011-09-08T19:42:26Z <p>This seems to be a question whose answer depends on whom you ask. Maybe we can come up with a final answer?</p> <p>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).</p> <p>What is unclear is whether Poincaré gave a correct <em>proof</em>. Poincaré's proof appears in two places: his paper</p> <p>Henri Poincaré, <em>Sur les groupes continus</em>, Transactions of the Cambridge Philosophical Society, 18 (1900), pp. 220–255 = Œuvres de Henri Poincaré, vol., III, Paris: Gauthier-Villars (1934), pp. 173–212,</p> <p>and the modern paper</p> <p><a href="http://smf.emath.fr/Publications/RevueHistoireMath/5/pdf/smf_rhm_5_249-284.pdf" rel="nofollow">T. Ton-That, T.-D. Tran, <em>Poincaré's proof of the so-called Birkhoff-Witt theorem</em>, Rev. Histoire Math., 5 (1999), pp. 249-284, also arXiv:math/9908139</a>.</p> <p>I have no access to the former source, so all my knowledge of the proof comes from the latter.</p> <p>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 <a href="http://smf.emath.fr/Publications/RevueHistoireMath/5/pdf/smf_rhm_5_249-284.pdf" rel="nofollow">the paper by Ton-That and Tran</a> 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).</p> <p>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 <a href="http://smf.emath.fr/Publications/RevueHistoireMath/5/pdf/smf_rhm_5_249-284.pdf" rel="nofollow">Ton-That and Tran paper</a>. My troubles lie within this paragraph on pages 277-278:</p> <p>"The first four chains are of the form</p> <p>\$U_1 = XH_1,\ U'_1 = H'_1Z,\ U_2 = YH_2,\ U'_2 = H'_2T\$,</p> <p>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\$."</p> <p><strong>Question:</strong> 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.</p> <p>What am I missing? Is this a mistake in Poincaré 1900? Or have the authors of <a href="http://smf.emath.fr/Publications/RevueHistoireMath/5/pdf/smf_rhm_5_249-284.pdf" rel="nofollow">1</a> misrepresented Poincaré's argument? Has anybody else tried to decipher Poincaré's proof?</p> <p><strong>PS.</strong> I have asked more or less asked this question some months ago, but it was hidden in <a href="http://mathoverflow.net/questions/61954" rel="nofollow">another question</a> and did not receive any answer. Mailing the authors did not help either. So my last hope is a public discussion.</p> http://mathoverflow.net/questions/74923/so-did-poincare-prove-pbw-or-not/74926#74926 Answer by Abdelmalek Abdesselam for So, did Poincaré prove PBW or not? Abdelmalek Abdesselam 2011-09-08T19:42:26Z 2011-09-08T19:42:26Z <p>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 <a href="http://www.digizeitschriften.de/main/dms/img/?PPN=GDZPPN002252635" rel="nofollow">Capelli's paper</a> (in French).</p>