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Consider the standard quantum group $U_q (\mathfrak{sl}_2)$ over the field $\mathbb{C}(q)$ of rational functions (or over $\mathbb{C}$ if $q \in \mathbb{C}$ is not a root of unity), with the usual Hopf algebra structure. Does $U_q (\mathfrak{sl}_2)$ admit a quasi-triangular structure?

Please be mindful that I am asking about $U_q (\mathfrak{sl}_2)$ and not about the topological Hopf algebra $U_h (\mathfrak{sl}_2)$, which is indeed quasi-triangular by the seminal work of Drinfeld. The $R$-matrix in the latter does not belong to the former, so the question is whether there exists a well-defined $R$-matrix in $U_q (\mathfrak{sl}_2)$. If not, could you provide a reference where it is proven that it does not exist?

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$U_q(\mathfrak{sl}_2)$ is indeed not quasitriangular with the usual $R$-matrix. Instead it satisfies a weaker condition of being braided [1].

One can certainly show that the usual universal R-matrix does not give a braiding on all finite-dimensional $U_q$-modules. When $q$ is a $2N$th root of unity there are modules on which $E^N$ and $F^N$ act by nonzero scalars, and the action of the $R$-matrix will not converge on the tensor product of two of these.

I don't know how to prove that there is no possible universal $R$-matrix that will work; this would follow if you could prove it was essentially unique but I'm not sure how to show that. The reference [1] makes the claim that $U_q$ is not quasitriangular but does not give a proof.

[1] Reshetikhin, N., Quasitriangularity of quantum groups at roots of 1, Commun. Math. Phys. 170, No. 1, 79-99 (1995). ZBL0838.17009.

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