Consider the Baker-Campbell-Hausdorff formula $\Phi(X,Y)\in\mathbb{Q}\langle\!\langle X,Y\rangle\!\rangle$ in non-commutative variables. Define $X*Y:=\Phi(X,Y)$ and $[X,Y]=(-X)*(-Y)*X*Y$, and then (as usual) define for any vector

$\mathbf{e}=(e_1,\ldots,e_r)\in\mathbb{N}^r$ the repeated commutator

$$[X,Y]_{\mathbf{e}}:=[X,\underbrace{Y,\ldots,Y}_{e_1},\underbrace{X,\ldots,X}_{e_2},\ldots]$$ (here $[X_1,\ldots,X_r]$ is defined as $[[X_1,\ldots,X_{r-1}],X_r]$).

I think that there is an analogous of the BCH formula which $XY-YX$ in terms of the commutators $[X,Y]_\mathbf{e}$. That is, if for $\mathbf{e}=(e_1,\ldots,e_r)$ we define $\langle\mathbf{e}\rangle=e_1+\ldots+e_r$ then there exist rational numbers $t_\mathbf{e}$ for all $\mathbf{e}\in\mathbb{N}^r$ and for all $r$ such that if we put $v_n(X,Y)=\sum_{<\mathbf{e}>=n}t_\mathbf{e}[X,Y]_\mathbf{e}$ then

$$XY-YX=\sum_{n\in\mathbb{N}}v_n(X,Y)$$.

I would appreciate any reference about this.

`\langle x\rangle`

instead of`<x>`

to obtain $\langle x\rangle$ instead of $<x>$. The spacing of`<`

is quite different because it is interpreted as a relation. $\endgroup$ – Mariano Suárez-Álvarez Mar 9 '12 at 17:05`\mathbb Q\rangle\!\rangle X,Y\langle\!\langle`

which gives $\mathbb Q\rangle\!\rangle X,Y\langle\!\langle$;`\newcommand`

is your friend :) ) $\endgroup$ – Mariano Suárez-Álvarez Mar 9 '12 at 17:08