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# Classifying irreducible finite-dimensional representations of the $q$-Weyl algebra

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!