Let $V_N$ denote the $N$-dimensional representation of the quantum group $U_q(\mathfrak s\mathfrak l_2)$. I am told that in the limit $N\to\infty$ with $q=e^{2\pi i/n}$ and $N/n\to\alpha\in(0,1)$, the representation $V_N$ of $U_q(\mathfrak s\mathfrak l_2)$ "converges" in some precise sense to an infinite-dimensional representation of the complex Lie group $\operatorname{SL}_2(\mathbb C)$.

What is the precise formulation of this result, and where is it discussed in the literature? [My searches so far have not found anything substantial.]