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where $V_{1,N-1}$ is the $N$-dimensional, $U_q\big(\mathfrak{sl}(2)\big)$-module of highest weight $\lambda=q^{N-1}$ and the $V\big(q^{n\alpha}\big)$ is the $U_q\big(\mathfrak{sl}(2)\big)$-Verma module of highest weight $\lambda=q^{n\alpha}=e^{2\pi i\alpha}$.
The $U_q\big(\mathfrak{sl}(2)\big)$-action for the $V_{1,N-1}$ module is given by: $$ K\cdot v_p=q^{N-2p-1}v_p, \ \ E\cdot v_p=[N-p]_qv_{p-1}, \ \ F\cdot v_p=[p+1]_qv_{p+1} $$ whereas its $N/n\to\alpha$ limit, gives the $U_q\big(\mathfrak{sl}(2)\big)$-action for the $V\big(q^{n\alpha}\big)$-Verma module: $$ K\cdot v_p=q^{n\alpha-2p-1}v_p, \ \ E\cdot v_p=[n\alpha-p]_qv_{p-1}, \ \ F\cdot v_p=[p+1]_qv_{p+1} $$ where $[x]_q=\frac{q^x-q^{-x}}{q-q^{-1}}$, as usual.

where $V_{1,N-1}$ is the $N$-dimensional, irreducible, $U_q\big(\mathfrak{sl}(2)\big)$-module of highest weight $\lambda=q^{N-1}$ and the $V\big(q^{n\alpha}\big)$ is the $U_q\big(\mathfrak{sl}(2)\big)$-Verma module of highest weight $\lambda=q^{n\alpha}=e^{2\pi i\alpha}$.
The $U_q\big(\mathfrak{sl}(2)\big)$-action for the $V_{1,N-1}$ module is given by: $$ K\cdot v_p=q^{N-2p-1}v_p, \ \ E\cdot v_p=[N-p]_qv_{p-1}, \ \ F\cdot v_p=[p+1]_qv_{p+1} $$ whereas its $N/n\to\alpha$ limit, gives the $U_q\big(\mathfrak{sl}(2)\big)$-action for the $V\big(q^{n\alpha}\big)$-Verma module: $$ K\cdot v_p=q^{n\alpha-2p-1}v_p, \ \ E\cdot v_p=[n\alpha-p]_qv_{p-1}, \ \ F\cdot v_p=[p+1]_qv_{p+1} $$ where $[x]_q=\frac{q^x-q^{-x}}{q-q^{-1}}$, as usual.

Let us now attempt to compute these limits; i will not include details on the handling of the order of the limits $N/n\to\alpha$, $n\to\infty$, since i admit i have not -yet- been able to obtain a rigorous formulation of the following but here is what i have got (mostly on an intuitive level):

In any case, if there is not some silly mistake in my conjecture (once more: the result comes from a mixture of computations and intuition), we finally arrive at an infinite dimensional $U\big(\mathfrak{sl}(2)\big)$-Verma module and from there we can get the corresponding infinite dimensional, $SL(2,\mathbb{C})$ representation. I will try to come back if -and when- i will be able to obtain a somewhat more rigorous formulation of the last limit.

Let us now attempt to compute these limits; i will not include details on the handling of the order of the limits $N/n\to\alpha$, $n\to\infty$, since i admit i have not -yet- been able to obtain a rigorous formulation of the following but here is what i have got:

In any case, if there is not some silly mistake in my conjecture (the result comes from a mixture of computations and intuition), we finally arrive at an infinite dimensional $U\big(\mathfrak{sl}(2)\big)$-Verma module and from there we can get the corresponding infinite dimensional, $SL(2,\mathbb{C})$ representation. I will try to come back if -and when- i will be able to obtain a somewhat more rigorous formulation of the last limit.

The "convergence" of the representation: Let us study the limit on the irreducible, f.d. representations of $U_{q}(\mathfrak{sl}_{2})$ with $q=e^{2\pi i/n}$:
(First recall that if $q$ is not a root of unity, then the f.d., irred, are parameterized by $\varepsilon=\pm 1$ and the positive integers, i.e. we will denote such $N$-dim, irred, modules as $V_{\varepsilon, N-1}$. The notion of the $N\to\infty$ limit here is well defined in the following sense: The $V_{1,N-1}$ modules here have matrix elements which are not generally continuous functions of $q$ but they can be handled with methods similar to those mentioned above to show that

The "convergence" of the representation: Let us study the limit on the irreducible, f.d. representations of $U_{q}(\mathfrak{sl}_{2})$ with $q=e^{2\pi i/n}$:
(First recall that if $q$ is not a root of unity, then the f.d., irred, are highest weight representations. They are parameterized by $\varepsilon=\pm 1$ and the positive integers, i.e. we will denote such $N$-dim, irred, modules as $V_{\varepsilon, N-1}$. The notion of the $N\to\infty$ limit here is well defined in the following sense: The $V_{1,N-1}$ modules here have matrix elements which are not generally continuous functions of $q$ but they can be handled with methods similar to those mentioned above to show that