I'm looking at page 4 and 5 of https://arxiv.org/pdf/math-ph/0105002.pdf. I am wanting to use (14) and (17) to verify the covariance of (14). But I'm having trouble, in particular, I'm having trouble proving $\frac{\partial}{\partial X'} Y' = q Y' \frac{\partial}{\partial X'}$. Here is what I have calculated:
\begin{align} \frac{\partial}{\partial X'} Y' & = \left( D \frac{\partial}{\partial X} - q C \frac{\partial}{\partial Y} \right)(C X + DY) \\ & = DC \frac{\partial}{\partial X} X + D^2 \frac{\partial}{\partial X} Y - q C^2 \frac{\partial}{\partial Y} X - q CD \frac{\partial}{\partial Y} Y \\ & = DC \left( q^2 X \frac{\partial}{\partial X} + 1 + (q^2 - 1) Y \frac{\partial}{\partial Y} \right) + q D^2 Y \frac{\partial}{\partial X} - q^2 C^2 X \frac{\partial}{\partial Y} - q CD \left( 1 + q^2 Y \frac{\partial}{\partial Y} \right) \\ & = q CD X \frac{\partial}{\partial X} - q^2 C^2 X \frac{\partial}{\partial Y} + q D^2 Y \frac{\partial}{\partial X} + DC \left( 1 + (q^2 - 1) Y \frac{\partial}{\partial Y} - q^2 - q^4 Y \frac{\partial}{\partial Y} \right) \end{align}
This doesn't seem to match $q Y' \frac{\partial}{\partial X'}$:
\begin{align} q Y' \frac{\partial}{\partial X'} & = q (CX + DY) \left( D \frac{\partial}{\partial X} - q C \frac{\partial}{\partial Y} \right) \\ & = q CD X \frac{\partial}{\partial X} - q^2 C^2 X \frac{\partial}{\partial Y} + q D^2 Y \frac{\partial}{\partial X} - q^2 DC Y \frac{\partial}{\partial Y} \end{align}
Where have I gone wrong?
