# $h$-adic Completion of $U_q(\frak{sl}_2)$?

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 $q$.

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Second Cohomology of semisinple lie alg vanishes. So any deformation is trivial. So the two algs are isomorphic. –  Alexander Chervov Mar 17 '13 at 11:49
Alexander> But $U_q$ is not a deformation of $U$, even if $q$ is a variable. Abtan> I'm confused about the objects you're considering: your $U_{\hbar}(\mathfrak g)$ is just the trivial deformation of $U(\mathfrak g)$, and not what people usually mean when they write $U_{\hbar}(\mathfrak g)$ (though they are indeed non canonically isomorphic as algebras). And if I understand correctly the second algebra you look at is $U_q(\mathfrak g)[[\hbar]]$ for some complex number $q$, and in particular without any relation between $q$ and $\hbar$, is it really what you meant ? –  Adrien Mar 17 '13 at 16:20
@Adrien: Sorry, the question was very badly written, it should be better now. –  Abtan Massini Mar 17 '13 at 16:34
Sorry problem with the editing - should be now ok. –  Abtan Massini Mar 17 '13 at 17:44