Understanding the notation to directly calculate $e^{ad_{X}}Y$

Question

I would appreciate help understanding the notation $e^{ad_X}(Y)$.

The notes I'm studying give several pieces, I just don't know how to put them together:

$e^{ad_X}= Ad(e^X)$ where $Ad(e^X)$ is conjugation by $e^X$, i.e., $Ad(e^X)Y=e^{X}Ye^{-X}$.

And it is proved that $ad_{X}Y=[X,Y]$, the Lie bracket.

But I don't know how to directly calculate $e^{ad_{X}}Y$ on its own, just using $ad_X$ in the exponent.

Thanks.

Answer 1

The crucial point is that $ad_X^n(Y)=[X,ad_X^{n-1}(Y)]$.

So $ad_X^2(Y)=[X,[X,Y]]=X^2Y-2XYX+YX^2$, etc.

Actually, $$ad_X^n(Y)=\sum_{k=0}^n {n\choose k} (-X)^kYX^{n-k} $$

Then by writing $$e^{ad_X}(Y)=\sum_{n\ge 0}\frac{1}{n!}ad_X^n(Y)$$ and doing some simple manipulations you can see that it equals $e^{X}Ye^{-X}$.