Representation of Lorentz algebra

[naima]

i find here a representation of the Lorentz algebra.
Starting from the matrix representation (with the $\lambda$ parameter) i see
how one gets the matrix form of $iJ_z$
I am less comfortable with the $-i y\partial_x + x \partial_y$ notation
Where does it come from? They say that it is a killing vector on $R^4$
I suppose that this is basic but as i read it in a physics paper i have not the complete background.

 

[fzero]

Let's look at an infinitesimal rotation around the $z$-axis, by a positive infinitesimal angle $\alpha$. We can just draw a diagram of the x-y plane and rotate the axes a bit counter clockwise. Then we'll see that
$$\begin{split}
& x \rightarrow x' = x + \alpha y, \\
& y \rightarrow y' = y - \alpha x, \\
& z \rightarrow z' = z. \end{split}$$
Now check that
$$\begin{split}
& - \alpha ( x \partial_y - y \partial_x) x = \alpha y, \\
& - \alpha ( x \partial_y - y \partial_x) y = - \alpha x. \end{split}$$
Therefore
$$ \delta x^k = i \alpha J_z x^k$$
if we define
$$ J_z = i ( x \partial_y - y \partial_x).$$

Further, we can show that $J_z$ is a Killing vector for the Minkowski metric. The Killing condition is
$$ (J_z)_{\mu ; \nu} + (J_z)_{\nu ; \mu} =0.$$
Since the Minkowski metric is flat, the covariant derivatives reduce to ordinary derivatives and the Killing condition becomes:
$$0 = (J_z)_{x ,y} + (J_z)_{y,x} = \partial_y ( -i y) + \partial_x ( i x),$$
which is obviously satisfied.

One can derive the rest of the generators by examining the other infinitesimal rotations and boosts and then verifying that the Killing equation is satisfied as I did above. Or else one can just compute the rest of the Killing vectors and then see that they have an interpretations as generators.