I take the following from Klimyk and Schmudgen's book Quantum Groups Page 167:

Proposition For $\frak{g}$ a complex semi-simple Lie algebra, there is an isomorphism $\phi$ of the $h$-adic algebra $U_{h}(\frak{g})$ onto $U({\frak g})[[h]]$ which coincides with the identity map modulo $h$.

Explicitly, for $U_h(\frak{sl}_2)$, the $h$-adic algebra generated by the elements $H,E$, and $F$, such that $$ [H,E] = 2E, ~~~~~ [H,F] = -2F, ~~~ [E,F] = (e^{h H} - e^{-hH})/(e^h - e^{-h}). $$ The isomorphism $\phi$ of the algebras is uniquely determined by its action on the generating elements and is given by $$ \phi(H) = H' ~~~~~ \phi(F) = F', ~~~~~ \phi(E) = 2\Big(\frac{\cosh h(H'-1) - \cosh 2h\sqrt{C'}}{|H'-1|^2-4C' \sinh^2 h}\Big), $$ where $H', E',F'$ are the generators of $U(\frak{sl}_2)$ satisfying the relations $[H',E'] = 2E'$, $[H',F'] = -2F'$, $[E',F'] = H'$, and $C' = \frac{1}{4}(H'-1)^2 + E'F'$ is the Casimir element of $U_h(\frak{sl}_2)$.

I would like to ask if anyone knows of an explicit description of this isomorphism for the case of $\frak{sl}_3$? The reference for the proof of the proposition is Drinfeld's 1986 ICM talk, or Shnider and Sternberg Quantum Groups Chapter 11.



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