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 […]

- Tags $\mu$ and $\lambda^m$ respectively. I now need to show that the isomorphism class of $V$ is determined by the scalars that $x^m, 1$ entry. Now I am trying to figure out how else to use the basis to classify $W$, any guide would be great!, but am not sure where to go from here. It seems like there are many possibilities, 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, dots, i, I can write the matrix of $y$ as $\text{diag}[\lambda, I have shown that $\{v, I have that $A = \langle x, j \in \mathbb{Z}$. The base field is $\mathbb{C}$. I need to classify the irreducible finite-dimensional representations. Assuming $V$ is su, q^{m-1}\lambda]$. I can also write the matrix of $x$ as having nonzero entries on the subdiagonal and the top right $m, q\lambda, 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 $V \cong W$, where $v$ is a $\lambda$-eigenvector of $y$. I have also shown (using Schur's lemma) that $x^m, with basis $x^iy^j$, x^{m-1}v\}$ is a basis, xv, y : yx = qxy \rangle$ is the $q$-Weyl algebra, y^m$ act by scalars, 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