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I'm looking at the article Baker–Campbell–Hausdorff formula - Wikipedia and I have a few questions.

Under the "Special cases" section, there is a notation $\DeclareMathOperator{\ad}{ad}$

$$ \log(e^X e^Y) = X + \frac{\ad_X}{1 - e^{-\ad_X}}Y + O(Y^2),$$

but it doesn't provide a reference for this formula. Who discovered this representation of the BCH formula and does it hold in the general case for general $X$ and $Y$ or is this only valid in the case $[X,Y] = sY$?

Another question related to this, is there a name for when $[X,Y] = sY$?

A third question is, what is the proper formal Taylor expansion for the $\frac{\ad_X}{1-e^{-\ad_X}}$ term? Is this convergence of this formal Taylor expansion limited by any particular spectral radius? The wiki article does not seem to go into detail about that.

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    $\begingroup$ There is a nice discussion of this, not answering your questions perhaps, on terrytao.wordpress.com/tag/baker-campbell-hausdorff-formula $\endgroup$
    – Ben McKay
    Commented Sep 26, 2022 at 15:38
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    $\begingroup$ @BenMcKay, re, there are three articles in that tag; did you mean Tao - The $C^{1, 1}$ Baker–Campbell–Hausdorff formula? $\endgroup$
    – LSpice
    Commented Sep 26, 2022 at 16:21
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    $\begingroup$ I don't know any name for it, but the situation $[X, Y] = s Y$ is precisely the case when there is a representation of the Lie algebra of the so called “$a x + b$ group” $\left\{\begin{pmatrix} a & b \\ 0 & 1 \end{pmatrix}\right\}$ sending $\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$ to $X$ and $\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ to $Y$ … so perhaps that could be used to make up a name? $\endgroup$
    – LSpice
    Commented Sep 26, 2022 at 16:24
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    $\begingroup$ See also the 2020 paper "Notes on the theorem of Baker-Campbell-Hausdorff-Dynkin" by Michael Mueger (math.ru.nl/~mueger/PDF/BCHD.pdf). $\endgroup$
    – Tom Copeland
    Commented Sep 26, 2022 at 18:32
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    $\begingroup$ And w.r.t. numerical integration of diff eqs, see the discussion surrounding eqns. 2.45 and 3.2 on pp. 33 and 38 of "Lie-group methods" by Iserles, Munthe-Kaas, Nørsett, and Zanna (damtp.cam.ac.uk/user/na/NA_papers/NA2000_03.pdf). $\endgroup$
    – Tom Copeland
    Commented Sep 26, 2022 at 18:56

1 Answer 1

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$\DeclareMathOperator\ad{ad}$The identity $$\log(e^X e^Y) = X + \frac{\ad_X}{1 - e^{-\ad_X}}Y + O(Y^2)\tag{$*$}\label{star}$$ is due to Poincaré [1], who showed that $W(t)=\log (e^X e^{tY})$ solves the ODE $$W'(t)=\frac{\ad_{W(t)}}{1-e^{-\ad_{W(t)}}}Y,\;\;W(0)=X.$$ The identity \eqref{star} is the solution to first order in $t$. It holds generally, you don't need $[X,Y]=sY$.

[1] H. Poincaré, Sur les groupes continus, Trans. Cambridge Phil. Soc., 18, 220–255 (1900). JFM 31.0386.01

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