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Here is a question:
Given a Lie algebra $\mathfrak{g} \simeq \big\langle (T_i)_{i=1}^n \big\rangle$ with bracket $[T_j, T_k] = f^i_{j k} T_i$, then alongside the BCH formula one can expand the Maurer-Cartan forms (left-invariant $(e^i)_{i = 1}^n$ with $\mathrm{d}e^i = -\tfrac{1}{2}f^i_{j k} \, e^j e^k$) of the corresponding Lie group in these coordinates as
$e^i \;=\; \mathrm{d}x^i - \tfrac{1}{2}f^i_{j k} x^j \, \mathrm{d}x^k + \tfrac{1}{6} f^i_{j k'} f^{k'}_{k l} x^j x^k \, \mathrm{d}x^l + \cdots \,.$Is this BCH-like series-expansion for the MC-forms citable from any source, for the first few terms?
Urs 2: it is an easy exercise (I had to do it 18 years ago) that the inverse of the MC matrix is the expression for the say left invariant vector field in terms of the same coordinate system on a neighborhood of unit element on a Lie group; in exponential coordinates – for example – one gets the linear part of the BCH formula, as it is clarified in our paper with Durov.
Chapter 7 has a conceptual intro, chap 2-6 are brute force calculational approach.
Here is the reasoning for the explicit formula in exponential coordinates, but a fresher version not mentioning MC is chapter 2 in LMP paper on Hopf algebroids,
Given any basis $e_1,\ldots,e_n$ of the tangent Lie algebra $T_e G$, extend it to a basis $e_1^*, \ldots,e_n^*$ of the space of left invariant vector fields on $G$, and consider the dual differential forms $\omega^1,\ldots,\omega^n$, that is $\omega^i(e_j) = \delta^i_j$ where $\delta^i_j$ is the usual Kronecker delta times the unit constant function on group $G$. Forms $\omega^i$ are sometimes called Maurer-Cartan forms, but more often nowdays the left-invariant $\mathfrak{g}$-valued form $\sum_{i=1}^n \omega^i e_i$, comprising all $\omega^i$-s, bears the same name.
$\exp : \mathfrak{g} \to G$ is a $C^\infty$-map, and it is a diffeomorphism when restricted to an open star-shaped neighborhood $U$ of $0\in \mathfrak{g}$. In particular, we may consider the pullbakcs $\exp^* \omega^i$ and hence define functions $A^i_j : X \mapsto (\exp^* \omega^i)_X (\partial'_j)$ where $X = \sum_k x^k e_k \in \mathfrak{g}$ and $\partial'_j$ corresponds to $e_j$ under the natural isomorpshism between $T_e G$ and in $T_X T_e G$. Denote also $\partial_j := d (\exp_X(\partial'_j)) \in T_{\exp{X}}$. Helgason shows~(Helgason, II 7, page 139), that for all $1\leq j\leq n$,
$\frac{1 - e^{-\ad{X}}}{\ad{X}} (e_j) = \sum_{i=1}^n A^i_j(X)\, e_i,$where $\ad{X}(X_j) = \sum_k (\sum_i x^i c_{ij}^k) e_k$ and the structure constants $c^k_{ij}$ in symbolic are defined by $[e_i, e_j] = c^k_{ij} e_k$. Thus $\ad{X}^N (e_j) = (m^N)^k_j e_k$, where $m^k_j := \sum_i x_i c^k_{ij}$ are the components of some matrix $m$, and $m^N$ is its $N$-th matrix power.
Let $A^{-1}$ be the inverse matrix of $A$, that is, $\sum_k A^i_j (A^{-1})^j_k = \delta^i_k$ etc. Let $e_l^* = \sum_j \chi_l^j \partial_j$. Then $(\omega^i)_{\exp{X}} ( \sum_j \chi_l^j \partial_j) = \delta^i_j$, hence $\sum_j \chi_l^j \omega^i (\partial_j) = \sum_j \chi_l^j A^i_j = \delta^i_l$ i.e. $\chi^r_l = (A^{-1})^r_l$. This yields our formula.
In general coordinates, one takes the MC matrix and calculates the equation for the inverse and then one compares it to the differential equation for function $\phi$ in our paper with Durov (the equation is in the chapter 4). The equation was btw in the very first letter of Lie around 1871 in which he actually proposed the idea of Lie algebra by looking at the differential equations for infinitesimal action up to second order. In the modern terms, the differential equation and the letter have been explained in short in the historical appendix to Lie groups book of Bourbaki written by Dieudonne.
Maybe you could look at Helgason and see that your formula is there implicitly.
II 7, page 139 in
Take into account the inverse matrix (and make sure if you work with the same convention left vs right invariant diff forms) to have the agreement in signs and coefficients.
BCH-like series-expansion for the MC-forms
It is from only the linear terms of BCH in fact, that is those which are linear in one of the two variables. Or, in differential geometric terms, it is from the formula for the differential of the exponential map (hence similarity to the Todd genus etc. in our case).
If one takes the Maurer-Cartan equation in ANY coordinates (not the normal coordinates induced by the exponential map) and uses the formula for the differential of the inverse matrix, and if the $\phi^i_j$ are the entries of the inverse matrix you get the differential equation for the components of the vector field. That is if the left invariant vector fields are
$X_j = \sum \phi^i_j(x) \partial_i$then $(A^{-1})^i_j$ satisfies the equation for $\phi^i_j$ iff $A$ satisfies MC. This is the exercise I was talking about.
The equation for $\phi^i_j$ is
$(\delta_\rho \phi^\gamma_\mu)\phi^\rho_\nu - (\delta_\rho \phi^\gamma_\nu)\phi^\rho_\mu = C^\sigma_{\mu\nu} \phi^\gamma_\sigma$where $C$ are the structure constants and $\delta_\rho$ is the $\rho$-th partial derivative.
Our approach in the paper with Durov is having diff. operators acting from the right and then using the antiautomorphism of the Weyl algebra so some signs and orders are opposite to the commutative viewpoint. This is in order to look at $X_i$ as coordinates in the noncommutative deformation.
The answer is: Helgason (2001), II Thm. 7.4.
Thanks!
Concretely:
That theorem gives the MC forms at $X = x^i T_i$ as
$e^i \;=\; \mathrm{d}x^i \left( \frac {1 - exp(- ad X)} {ad X} (\partial_k) \right) \mathrm{d}x^k$where (by ibid, p. 36)
$\frac{1 - exp(-A)}{ A } \;\; \coloneqq \;\; \sum_{n=0}^\infty \tfrac{1}{(n+1)!} (-A)^n \,,$so that:
$e^i \;=\; \mathrm{d}x^i \left( \textstyle{ \sum_{n = 0}^\infty } \tfrac{1}{(n+1)!} ( - ad X )^n (\partial_k) \right) \mathrm{d}x^k \,.$Unwinding this with
$(ad X) \;=\; \big( x^j f^\bullet_{j \bullet} \big)$we get my little formula from #2
$e^i \;=\; \tfrac{1}{1!} \mathrm{d}x^i - \tfrac{1}{2!} f^i_{jk}x^j \mathrm{d}x^k + \tfrac{1}{3!} f^i_{j k'} f^{k'}_{k l} x^j x^k \mathrm{d}x^l + \cdots$Will be adding this now to Maurer-Cartan form.
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