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I'm looking for list of identities involving adjoint action $\mathrm{ad}_A X = [A,X] = AX - XA$. For example, it can be easily shown that:

\begin{equation} e^{\mathrm{ad}_A} X = e^A X e^{-A} \end{equation}

Are there any others listed somewhere or can anybody provide some?

Currently, I'm mostly interested in when is it possible to define something like $\mathrm{ad}_A^{-1}$ in a sense that $[A,\mathrm{ad}_A^{-1}X]=X$ and especially interested in finding closed form for \begin{equation} \mathrm{ad}_A^{-1}(e^{\mathrm{ad}_A}-1) \equiv \sum_{n=0}^\infty\frac{\mathrm{ad}_A^n}{(n+1)!} \end{equation}but other identities involving adjoint action might come in handy as well.

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1 Answer 1

A useful identity is $ad_{D(x)}=[D,ad_x]$ for every derivation $D$. Furthermore, the Baker-Campell-Hausdorff formula is given by $$ \begin{align} Z(x,y) & =\log \exp(x)\exp(y)\\ & =x+y+\frac{1}{2}ad_x(y)+\frac{1}{12}ad_x^2(y)+\frac{1}{12}ad_y^2(x)\\ & -\frac{1}{24}ad_y ad_x^2(y)+\frac{1}{720}ad_y^4 (x)\pm \cdots \end{align} $$

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