Timeline for Where did Sophus Lie write the group commutator for two one parameter groups
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15 events
| when toggle format | what | by | license | comment | |
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| Sep 19, 2016 at 21:21 | comment | added | Danu | @RobertBryant I think it would be very useful to work all the material of the comments into the actual answer, if you could find the time. | |
| Jun 25, 2013 at 13:22 | comment | added | Robert Bryant | Note that in [1891], Section 8, Lie specifically says that 'infinitesimal transformations' are special cases of 'infinitely small transformations'; indeed, the former are infinitely small transformations that form a $1$-parameter subgroup. Thus, the $S$ and $T$ would be infinitesimal transformations, but the commutator itself need not be. There is the added subtlety in Lie's terminology that 'infinitely small transformations' are essentially 'finite-parameter transformations' for which the parameters are 'infinitesimal'. I think that the modern equivalent would be 'germs of transformations'. | |
| Jun 24, 2013 at 3:34 | comment | added | Francois Ziegler | Extra data point: the very same development of [1883,1884,1891] (including the expression $S^{−1}T^{−1}ST$) is found once more on p. 577 of S. Lie & F. Engel [1893]: Theorie der Transformationsgruppen, vol. 3 (archive.org/stream/theotransformation03liesrich#page/n610). I believe that all transformations ($x_i\mapsto x_i''$, $S$, $T$,...) in this discussion are described as "infinitesimal". | |
| Jun 24, 2013 at 3:32 | comment | added | Francois Ziegler | If we take Peter's title question literally ("Where did Sophus Lie write the group commutator...?"), then it does make a difference whether he not only "considered" the commutator, but positively did, himself, write things like $T^{−1}S^{−1}TS$ (which to me "smell" more like Engel; compare e.g. archive.org/stream/berichteberdiev35klasgoog#page/n567). Likewise for the "Trotter" formula. Also, I may be wrong but I read [1883,1884] as composing/commuting "infinitesimal", not "infinitely small" transformations, no? | |
| Jun 24, 2013 at 3:12 | comment | added | Francois Ziegler | @Robert, regarding "all-important": I do agree that [1891]'s text between (44) and (45) describes the commutator in words. What I just now realize is that I was (probably mis-)taking your "first displayed equation in Paragraph 36, which is exactly the first formula you asked about" to be (46), the displayed $T^{−1}S^{−1}TS$. (It's the first displayed item, and is your answer to Peter's title question; but indeed it's not an equation like the next one.) So I thought it was all-important to you, too! | |
| Jun 22, 2013 at 17:41 | comment | added | Robert Bryant | @Francois: Also, I'm not convinced that the writing out of the symbolic expression $T^{-1}S^{-1}TS$, as Lie (or, possibly, as you point out, Engel) does in [1891] is 'all-important'. It seems clear, even in [1883], that Lie is computing the ('infinitely small', in his sense) 'difference' between the transformations $ST$ and $TS$ because he says, in words, that that is what he is doing—finding the 'infinitely small' transformation that relates these two compositions. Is it so important that he explicitly calls this difference a 'commutator' or that he writes it out symbolically? | |
| Jun 22, 2013 at 14:24 | comment | added | Robert Bryant | @Francois: Thanks for these comments; I share your concerns that we current mathematicians often read our own modern understanding into historical documents. What's interesting in this case is that the [1883,1884,1891] calculations that Lie did with 'infinitesimals' to get these formulae are identical to the calculations that you and I would do (using the chain rule, Taylor series, and existence/uniqueness of ODE with parameters for our justification), to derive the Taylor series (in the flow parameters) of the commutator of the two flows. So what's the real difference, the names we use? | |
| Jun 21, 2013 at 21:27 | comment | added | Francois Ziegler | Now to thicken the plot: If you read [1891] carefully, you'll see that it concludes with a statement (completely mistranslated in the neo-classical-physics.info version): "38. The foregoing investigation was, like the one on linear differential equations, worked out by Professor Engel after a manuscript of mine." Moreover Paragraph 36 (containing the all-important expression $T^{−1}S^{−1}TS$) is among those in small type. Could that mean that it's in fact an elaboration due to Engel? | |
| Jun 21, 2013 at 21:03 | comment | added | Francois Ziegler | That's ok :-) Whether one decides that your formula "really is" in Lie may ultimately hinge on a distinction famously drawn by I. Grattan-Guinness (ams.org/mathscinet-getitem?mr=2026308), between the view of the past by "inheritors" (who like to emphasize the similarities), versus "historians" (who like to emphasize the differences). In any event, I believe that with [1883], [1884] and especially [1891], we have found the closest thing to an antecedent in Lie. | |
| Jun 21, 2013 at 13:18 | comment | added | Peter Michor | Many thanks. I am now traveling, and then our library is relocating and thus closed, so I will check all references when I can. Sorry, Francois, I changed the accepted answer to Robert's. | |
| Jun 21, 2013 at 13:16 | vote | accept | Peter Michor | ||
| Jun 21, 2013 at 13:16 | vote | accept | Peter Michor | ||
| Jun 21, 2013 at 13:16 | |||||
| Jun 20, 2013 at 23:25 | comment | added | Robert Bryant | Actually, one must be a little careful about Lie's terminology. Lie makes a distinction between infinitesmial transformations and infinitely small transformations in a group. The former we recognize as vector fields today, but the latter are better thought of as germs of (ordinary) finite-parameter families of transformations of the group near the identity. The parameters in such families are infinitely small only in the sense that one only uses information about them in an arbitrarily small neighborhood of the identity. See Paragraphs 7, 8 and 9 of the the 1891 paper I quoted above. | |
| Jun 20, 2013 at 19:51 | comment | added | Francois Ziegler | You're right, I overlooked that. And the very same formulas are indeed in [1883], as well as the [1884] proof I quoted. But key difference, in 1883-1884 Lie only claimed to be composing infinitesimal, not finite, transformations. (Indeed he described his parameters $\omega_i$, $\delta t$, $\delta\tau$ there as "infinitesimal", whereas in [1891] he has finite $\varepsilon$s.) [1883]: archive.org/stream/gesammabhan05lierich#page/n375 [1884]: archive.org/stream/gesammabhan06lierich#page/n144 [1891]: archive.org/stream/gesammabhan06lierich#page/n357 | |
| Jun 20, 2013 at 15:03 | history | answered | Robert Bryant | CC BY-SA 3.0 |