ad

  
Purpose 		

Implement the adjoint operator $(ad_X^n)$ applied $n$ times to $Y$, 
which corresponds to the $n$-times Lie bracketing of $X$ with $Y$.





Syntax 		
e:=ad(x,y,n);




Description 		

This procedure implements the adjoint operator $(ad_X^n)$ applied $n$
times to $Y$ and which corresponds to the $n$-times Lie bracketing of
$X$ with $Y$, i.e.

$$\begin{eqnarray*}
(ad_X^0)Y &=& (ad_X)^0Y=Y\\ 
(ad_X )Y &=& [X,Y]\\ 
(ad_X^n)Y &...
...=(ad_X^{n-1})[X,Y]\\ 
 &=& (ad_X)(ad_X^{n-1})Y=[x,(ad_X^{n-1})Y]
\end{eqnarray*}$$










Arguments 		
 $x, y$  $\parbox{0.64\textwidth}{A pair of elements of the
 Lie algebra or Lie polynomials.}$


 		 $n$ The number of times $x$ will be bracketed with $y$.




Example 		

The calculation of the Lie bracket $[f_1,[f_1,f_0]]=(ad_{f_1}^2)f_0$
can be performed as follows:
> ad(f1,f0,2);

                         f1~ &* (f1~ &* f0~)