Timeline for Nice proofs of the Poincaré–Birkhoff–Witt theorem
Current License: CC BY-SA 4.0
30 events
| when toggle format | what | by | license | comment | |
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| Oct 3, 2022 at 11:37 | history | edited | Martin Sleziak |
added the tag (pbw-theorems)
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| Mar 6, 2019 at 18:06 | history | edited | YCor | CC BY-SA 4.0 |
minor clarifications
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| Nov 28, 2018 at 9:41 | answer | added | Pedro | timeline score: 7 | |
| Nov 24, 2018 at 12:20 | answer | added | Claudio Gorodski | timeline score: 7 | |
| May 31, 2018 at 16:26 | comment | added | Duchamp Gérard H. E. | @GeraldEdgar In his paper "the diamond lemma for ring theory" (which reshows PBW theorem), George M. Bergman cites H. Poincaré, Sur les groupes continus, Trans. Cambridge Philos. Soc. 18 (1900), 220-255. I did not check yet. | |
| Feb 26, 2018 at 14:08 | comment | added | Julien Grivaux | @PeterMay You can find a proof in the book "Lie Superalgebras and Enveloping Algebras" by Ian M. Musson. | |
| May 7, 2017 at 9:59 | answer | added | user25309 | timeline score: 21 | |
| Feb 5, 2016 at 14:02 | answer | added | David E Speyer | timeline score: 10 | |
| Feb 8, 2013 at 0:20 | comment | added | Peter May | No, I was being stupid. The MM proof in characteristic zero is only marred by minor typos and is really slick. I'm still not very happy with the char p proof, although I guess it works. Let me ask again if there is another proof in the literature for restricted graded Lie algebras (i.e. Lie algebras plus a pth power operation). | |
| Feb 7, 2013 at 18:32 | comment | added | John Palmieri | Peter: don't B.3.8 and B.3.9 just refer to 3.8 and 3.9? Maybe I'm missing something, though. | |
| Feb 7, 2013 at 4:16 | comment | added | Peter May | Actually, rereading MM 50 years later, I do think I understand what they had in mind in the char p case, but it is unsettling: they use the char 0 case to prove the char p case for free Lie algebras and deduce the result from that. A direct proof feels more satisfactory. | |
| Feb 7, 2013 at 4:10 | comment | added | Peter May | What is more interesting is the PBW for restricted Lie algebras in finite characteristic. That is not mentioned in Milnor and Moore, but all known general structure theorems for graded connected Hopf algebras in char p can be derived from it. The proof in the restricted case and its application to the cited structure theorems are in ``More concise algebraic topology (Kate Ponto and myself) and I really am curious whether that version of the graded PBW theorem ever appeared anywhere else (I first proved it in the 1960's: John, you will see its relevance to my thesis.) | |
| Feb 7, 2013 at 3:59 | comment | added | Peter May | Deligne and Morgan say ``Our arguments used characteristic zero in an essential way'', which makes their argument uninteresting to an algebraic topologist. They give a reference to Corwin, Ne'eman and Sternberg for a more general proof that still excludes the slightly subtler cases of characteristic 2 or 3. John, there is a history of that part of MM I won't go into here. The char 0 case (p. 243) refers to missing B.3.8 and B.3.9 for the proof. The char p case (pp 251-2) is just 9 lines plus a little diagram, and I never could figure it out. (I'll add more in another comment.) | |
| Feb 6, 2013 at 18:09 | comment | added | John Palmieri | @Peter: what about the proof in Milnor and Moore? | |
| Feb 6, 2013 at 3:00 | comment | added | Todd Trimble | @Peter: does the reference to the Deligne and Morgan article I mentioned above fit the bill? It would sure seem to. | |
| Feb 6, 2013 at 2:59 | comment | added | Todd Trimble | From George Bergman's A Diamond Lemma for Ring Theory comes an reply to Birkhoff: H. POINCARE, Sur les groupes continus, Trans. Cambridge Philos. Soc. 18 (1900), 220-255. | |
| Feb 6, 2013 at 2:58 | comment | added | Peter May | This is to answer a question with a question. Does anyone know of a published proof, nice or not, of the PBW for graded Lie algebras, with the usual sign convention on the symmetry of the graded tensor product? Ponto and I put a proof in ``More Concise algebraic Topology'', Section 22.1, generalizing the passage to associated graded version of the classical proof. | |
| Feb 6, 2013 at 2:24 | answer | added | David Jordan | timeline score: 14 | |
| Feb 5, 2013 at 15:55 | answer | added | David E Speyer | timeline score: 7 | |
| Feb 5, 2013 at 15:42 | comment | added | Gerald Edgar | An aside: Birkhoff would sometimes remark that he didn't know what Poincaré had to do with the result. | |
| Feb 5, 2013 at 15:14 | comment | added | Todd Trimble | Adding to what Qiaochu wrote, PBW can be proved in the general setting of a $k$-linear tensor category, $k$ a field of characteristic zero. Such a streamlined categorical approach is especially helpful since, for example, it subsumes the pain involved with signs when one wants PBW in a super-setting. See chapter 1 of Notes on Supersymmetry by Deligne and Morgan, p. 48 ff., in Quantum Fields and Strings: A Course for Mathematicians, vol. I for a proof. | |
| Feb 11, 2012 at 4:11 | answer | added | Jim Conant | timeline score: 5 | |
| Feb 11, 2012 at 3:07 | answer | added | n m | timeline score: 4 | |
| Feb 3, 2012 at 12:19 | comment | added | Vladimir Dotsenko | @darij: yes, of course you are right. I guess I was more referring to the statement than to the proof. | |
| Feb 3, 2012 at 11:27 | comment | added | darij grinberg | I wouldn't call the Braverman/Gaitsgory proof a generalization of the diamond lemma argument, or did I fail to read between the lines in that proof? | |
| Feb 3, 2012 at 10:27 | comment | added | Vladimir Dotsenko | A small comment - the shortest and in my opinion nicest way to organise the algebraic manipulations you mention is to use Diamond Lemma, - and in this way can be generalised wonderfully, see e.g. arxiv.org/abs/hep-th/9411113. | |
| Feb 3, 2012 at 9:48 | answer | added | darij grinberg | timeline score: 29 | |
| Feb 3, 2012 at 8:25 | comment | added | DamienC | Since you are asking for a list then I guess this should be community wiki. | |
| Feb 3, 2012 at 6:03 | comment | added | Qiaochu Yuan | I don't know if this counts as a reformulation, but PBW can be interpreted as the statement that the associated graded of $U(\mathfrak{g})$ can be naturally identified with $S(\mathfrak{g})$; one should interpret the former as a noncommutative algebra of operators on the quantum system whose classical limit is the Poisson manifold $\mathfrak{g}^{\ast}$. | |
| Feb 3, 2012 at 5:41 | history | asked | user332 | CC BY-SA 3.0 |