I have that $A = \langle x, y : yx = qxy \rangle$ is the $q$-Weyl algebra, with basis $x^iy^j$, $i, j \in \mathbb{Z}$. The base field is $\mathbb{C}$.

I need to classify the irreducible finite-dimensional representations. Assuming $V$ is such a module, I have shown that $\{v, xv, \dots, x^{m-1}v\}$ is a basis, where $v$ is a $\lambda$-eigenvector of $y$. I have also shown (using Schur’s lemma) that $x^m, y^m$ act by scalars, $\mu$ and $\lambda^m$ respectively. I now need to show that the isomorphism class of $V$ is determined by the scalars that $x^m, y^m$ act by.

All I know is that another representation $W$ is isomorphic to $V$ if and only if there is an intertwining $\phi: V \to W$ and that $V \cong W$. Using the basis given above, I can write the matrix of $y$ as $\text{diag}[\lambda, q\lambda, \dots, q^{m-1}\lambda]$. I can also write the matrix of $x$ as having nonzero entries on the subdiagonal and the top right $m,1$ entry.

Now I am trying to figure out how else to use the basis to classify $W$, but am not sure where to go from here. It seems like there are many possibilities, since all that needs to be checked are that $x$ and $y$ act in the same way. Somehow I feel like I need to use that $V \cong W$, but cannot see how to proceed from there. I thought that I could write out an isomorphism $\phi$ using the basis of $V$ but all I can tell is that $\phi(x^kv)$ has to correspond to some other basis element of $W$. I am mostly lost from here, any guide would be great!