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For the algebra (I am for forgetting the Hopf structure) $U_q(\frak{sl}_2)$ defined over ${\mathbb C}$, and the formal power series version/ $h$-adic version $U_h(\frak{sl}_2)$, which I think as the polynomial algebra over $U(\frak{sl}_2)$ completed with respect to the usual $h$-adic metric. Now if I take the formal power series algebra of $U_q(\frak{sl}_2)$, ie the trivial deformation, then how will this relate to $U_h(\frak{sl}_2)$? It seems to me that they will be isomorphic, assuming one can somehow relate $h$ and $q$.

Consider the algebra (I am for forgetting the Hopf structure) $U_q(\frak{sl}_2)$ defined over ${\mathbb C}$, and the formal power series version/ $h$-adic version $U_h(\frak{sl}_2)$, which I think as the complex algebra freely by the elements $E,F,H$, completed with respect to the usual $h$-adic metric, and quotiented by the closure of the ideal generated by the usual set of generators. Now if I take the complex formal power series algebra of $U_q(\frak{sl}_2)$, ie the trivial deformation, then how will this relate to $U_h(\frak{sl}_2)$? It seems to me that they will be isomorphic, assuming one can somehow relate $h$ and $q$.

For the algebra (I am for forgetting the Hopf structure) $U_q(\frak{sl}_2)$ defined over ${\mathbb C}$, and the formal power series version/ $h$-adic version $U_h(\frak{sl}_2)$, which I think as the polynomial algebra over $U(\frak{sl}_2)$ completed with respect to the usual $h$-adic metric. Now if I take the formal power series algebra of $U_q(\frak{sl}_2)$, ie the completion of its polynomial algebra with respect to the usual $h$-adic metric, then how will this relate to $U_h(\frak{sl}_2)$? It seem to me that they will be isomorphic, but it is not 100% clear, for example the root of unity case should cause problems.

For the algebra (I am for forgetting the Hopf structure) $U_q(\frak{sl}_2)$ defined over ${\mathbb C}$, and the formal power series version/ $h$-adic version $U_h(\frak{sl}_2)$, which I think as the polynomial algebra over $U(\frak{sl}_2)$ completed with respect to the usual $h$-adic metric. Now if I take the formal power series algebra of $U_q(\frak{sl}_2)$, ie the trivial deformation, then how will this relate to $U_h(\frak{sl}_2)$? It seems to me that they will be isomorphic, assuming one can somehow relate $h$ and $q$.

For the algebra (I am for forgetting the Hopf structure) $U_q(\frak{sl}_2)$ defined over ${\mathbb C}$, and the formal power series version/ $h$-adic version $U_h(\frak{sl}_2)$, which I think as the polynomial algebra over $U(\frak{sl}_2)$ completed with respect to the usual $h$-adic metric. Now if I take the formal power series algebra of $U_q(\frak{sl}_2)$, ie the completion of its polynomial algebra with respect to the usual $h$-adic metric, then how will this relate to $U_h(\frak{sl}_2)$? It seem to me that they will be isomorphic, but it is not 100% clear.

For the algebra (I am for forgetting the Hopf structure) $U_q(\frak{sl}_2)$ defined over ${\mathbb C}$, and the formal power series version/ $h$-adic version $U_h(\frak{sl}_2)$, which I think as the polynomial algebra over $U(\frak{sl}_2)$ completed with respect to the usual $h$-adic metric. Now if I take the formal power series algebra of $U_q(\frak{sl}_2)$, ie the completion of its polynomial algebra with respect to the usual $h$-adic metric, then how will this relate to $U_h(\frak{sl}_2)$? It seem to me that they will be isomorphic, but it is not 100% clear, for example the root of unity case should cause problems.