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If $X,Y$ are vector fields and $\def\Fl{\operatorname{Fl}}\Fl^X_t$ and $\Fl^Y_t$ their local flows, let $[\Fl^X_t,\Fl^Y_t]:= \Fl^Y_{-t}\Fl^X_{-t}\Fl^Y_t\Fl^X_t$ denote the group commutator of the flows. Then
$$ \partial_t|_0 [\Fl^X_{t},\Fl^Y_{t}] =0, \qquad \frac12 \partial_t^2|_0 [\Fl^X_{t},\Fl^Y_{t}] = [X,Y]. $$ See

  • Markus Mauhart, Peter W. Michor: Commutators of flows and fields. Archivum Mathematicum (Brno) 28,3-4 (1992), 228–236, (pdf)

for an extension of this to formal bracket expressions of arbitrary length.

Where did Sophus Lie write this? And where did he compute something like $$ \lim_n \frac1n [\Fl^X_{1/n},\Fl^Y_{1/n}]^n\quad ? $$

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Great question! I have also been wondering about these for some time now. There are many publications of Lie available online but I could not find the reference. –  András Bátkai Jun 19 '13 at 12:27
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I don't have access to the collected works of Lie just now, but you might be able to find the earliest published reference by looking in Thomas Hawkins's book The Emergence of the Theory of Lie Groups: An Essay in the History of Mathematics (1869-1926). I believe that he discusses Lie's early work on transformation groups there, and there should be some mention of where Lie first wrote about the bracket and where he derived many of its properties. –  Robert Bryant Jun 19 '13 at 12:35
    
@Robert: Many thanks. I will try to find the book. –  Peter Michor Jun 19 '13 at 15:04
    
The Trotter product formula for the matrix (=finite dimensional) case should follow also from the Baker–Campbell–Hausdorff formula. –  András Bátkai Jun 19 '13 at 18:40

2 Answers 2

up vote 16 down vote accepted

Looking around on the internet, I found an English translation of Lie's 1891 paper Die Grundlagen für die Theorie der unendlichen kontinuierlichen Transformationsgruppen. I. (I.e., The foundations of the theory of infinite continuous transformation groups - I) at the location

http://neo-classical-physics.info/uploads/3/0/6/5/3065888/lie-_infinite_continuous_groups_-_i.pdf

In this English translation (I haven't gone to look for the original German), Lie gives the formula for the commutator of the flows of two vector fields in the form that you require. Look at Paragraphs 35-37, noting especially the displayed equation (45) and the first displayed equation in Paragraph 36, which is exactly the first formula you asked about.

It's quite possible that these formulae appeared even earlier in Lie's papers. He mentions, in the introduction, a paper of 1883 that might well also have these formulae. I'll check when I get the chance.

Concerning the Trotter formula (your second formula), I have no idea.

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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 –  Francois Ziegler Jun 20 '13 at 19:51
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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. –  Robert Bryant Jun 20 '13 at 23:25
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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. –  Francois Ziegler Jun 21 '13 at 21:03
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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? –  Francois Ziegler Jun 21 '13 at 21:27
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@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? –  Robert Bryant Jun 22 '13 at 14:24

I believe you're not going to find exactly what you want in Lie, because he never formalized flows (or finite transformations) and their commutation as you do. Maybe the closest would be this, from Über Differentialinvarianten, Math. Ann. 24 (1884) 537-578:

... erhalten wir folgenden Fundamentalsatz, den ich 1872 entdeckt habe:

Satz 3. Enthält eine kontinuierliche Gruppe die beiden infinitesimalen Transformationen: $$ Bf=\sum\xi_\varkappa\frac{\partial f}{\partial x_\varkappa} \quad\textit{und:}\quad Cf=\sum\eta_\varkappa\frac{\partial f}{\partial x_\varkappa}, $$ so enthält sie ebenfalls die infinitesimale Transformation: $$ \sum_i(B\eta_i-C\xi_i)\frac{\partial f}{\partial x_i}, $$ deren Symbol bekanntlich auf die beiden äquivalenten Formen: $$ B(C(f)) - C(B(f)) = (B, C) $$ gebracht werden kann.

As you can see, his definition of the bracket of vector fields is always as the commutator of the derivations they define on functions (something that goes back to Jacobi). What this Satz states, then, is that the finite transformations (or flow) generated by the infinitesimal commutator $(B,C)$ belong to the group generated by (the flows of) $B$ and $C$. Not surprisingly, Lie's proof is by expanding the flows to second order.

Lie may or may not have stated this Satz elsewhere before 1884, but I doubt he ever wrote a formula for, much less definition of, the bracket as limit of commutators of finite transformations.

Correction Robert Bryant has now found an 1891 reference where Lie (or at least Engel) indeed commutes finite transformations. See his reply and the comments there.

Update As to your question of who (esp. first) expressed the bracket as a derivative of commutators of flows: I don't know (my impression is that these things developed slowly in a sort of consensus). As a data point though, one might argue that the formula $$ [V,T]=\frac{d}{ds}\frac{d}{dt}e^{-sV}e^{tT}e^{sV}\Bigr|_{s=t=0} $$
is on p. 240 of Poincaré, Sur les groupes continus, Trans. Cambridge Philos. Soc. 18 (1900) 220-255.

Further update Trotter's formula that you also mention now is indeed called "Lie-Trotter" by e.g. Chernoff [1968,1974] or Chorin et al. [1978]. The latter write (sic):

... the equation $dx/dt=Ax+Bx$ leads to the 1875 formula of S. Lie [38]: $$ \exp\{A+B\} = \lim_{n\to\infty}(\exp\{A/n\}\exp\{B/n\})^n.\tag{$*$} $$ This and the related formula $$ \exp\{[A,B]\} = \lim_{n\to\infty}( \exp\bigl\{\frac{-B}{\sqrt n}\bigr\} \exp\bigl\{\frac{-A}{\sqrt n}\bigr\} \exp\bigl\{\frac{B}{\sqrt n}\bigr\} \exp\bigl\{\frac{A}{\sqrt n}\bigr\})\tag{$**$} $$ occur in the theory of Lie groups.

...

[38] Lie, S., and Engel, F., Theorie der Transformationsgruppen, 3 Vols., Teubner, Leipzig, 1888.

The problem is that [38] is not from 1875, nor does it contain anything remotely like formula ($*$) (I am ready to bet a lot of money). I may be wrong but until someone finds that elusive 1875 paper, I would tend to date ($*$) and ($**$) from around von Neumann [1929, p. 19].

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Many thanks. If not Lie, who wrote this formula for the first time? The name Trotter product formula come to my mind. –  Peter Michor Jun 19 '13 at 18:08
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I believe that Lie did, in fact, write down the explicit Taylor expansion for the commutator of the flows of two vector fields and tied it to the bracket. See my answer below. –  Robert Bryant Jun 20 '13 at 15:05

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