$\mathfrak{sl}(2)$ (over $\mathbb{C}$) with basis $E_\pm, H$ with commutation relations $$ [H,E_{\pm}]=\pm 2 E_\pm,\quad [E_+,E_-]=H $$ admits the well-known representation on $\mathbb{C}[x]$ with $$ E_+ = \partial_x,\quad E_- = -x^2 \partial_x + s\,x,\quad H = -x \partial_x - s $$ where $\partial_x = \frac{d}{dx}$. This representation is highest-weight with highest-weight vector given by $1\in \mathbb{C}[x]$. The parameter $s$ is free to take any value in $\mathbb{C}$. Furthermore this differential operator realisation can also be used if one changes the space of functions on which we act. For example, the principle series representations can be realised in this way.
Question: Can highest-weight representations of $U_q(\mathfrak{sl}(2))$ be realized in this way? I am aware that $U_q(\mathfrak{sl}(2))$ admits a representation on the quantum plane $\mathbb{C}_q[x,y]$ - this is not the representation I am looking for since that has a direct counterpart for usual $\mathfrak{sl}(2)$, which is not the representation I described at the beginning.
I am familiar with the textbooks by Chari and Pressley and by Kassel but have not come across such a representation.
I have tried to construct it directly, granted using a rather naive approach and substituting the differential operator $\partial_x$ with the $q$-differential operator $D_q$. This did not work due to the fact that the Liebniz rule for q-differentiation involves a multiplication by $q$ in the functions which are not differentiated. q-derivative
Does such a representation exist? If so please provide a reference - I am also interested in the generalisation to higher-rank cases.
Edit: The representation on $\mathbb{C}_q[x,y]$ is not the representation I am looking for as in the $q\rightarrow 1$ limit this reduces to a representation on $\mathbb{C}[x,y]$, not $\mathbb{C}[x]$ as I described in the initial paragraph of the question. The representation I am looking for should reduce to the one described initially for $q\rightarrow 1$.
