Reference request: "duality" relations between $U_q(\mathfrak{g})$, $O_q(G)$ and $O_q(G^*)$ Let $\mathfrak{g}$ be a bialgebra, $\mathfrak{g}^*$ its dual, and $G$ and $G^*$ the corresponding connected simply-connected Poisson-Lie groups. I have repeatedly heard claims of the following flavour, but never found a trace in any of the standard texts I have at hand:
1) The Hopf algebras $U_q(\mathfrak{g})$ and $O_q(G)$ are dual, generalising the classical pairing for $U(\mathfrak{g})$ and $O(G)$ (for which I'm also hoping to see a proof).
2) $U_q(\mathfrak{g})$ is isomorphic to $O_q(G^*)$, or at least contains the latter.
References/comments regarding the analogous story for $U_h(\mathfrak{g})$ would also be very useful.
 A: $\newcommand{\g}{\mathfrak{g}}$
This is known as the "quantum duality principle". FIrst of all, note that for arbitrary Lie bialgebra there"s no such things as $U_q(\g)$ in general, so everything works for formal deformations. If $\g$ is semisimple this can be turned into statements about the $q$ version. This is due to Drinfeld, and has been done rigorously by Gavarini (http://arxiv.org/abs/math/9909071).
He constructs equivalence of categories between the category of quantum envelopping algebra, and quantized formal power series algebras. Informally it maps $U_{\hbar}(\g)$ to the "subalgebra generated by $\hbar g$", denoted by $U_{\hbar}(\g)'$. The latter is then a flat deformation of $\widehat{S}(\g
)$, which you can identify with the completion of $O(G^*)$ at the identity. He also show the existence of a natural Hopf pairing between $U$ and $U'$. To the best of my knowledge, this is actually how you define what is also denoted by $O_{\hbar}(G^*)$ in the general case.
It's instructive to think about the Lie algebra case: $\g^*$ is then abelian, and the Lie bracket on $\g$ gives a Poisson structure on $\g^*$. Now in that case $U_{\hbar}(\g)=U(\g)[[\hbar]]$ is a deformation of the co-Poisson Hopf algebra $U(\g)$ with trivial co-Poisson structure, and $U(\hbar \g)[[\hbar]]$ is a deformation quantization of the Poisson structure on $G^*=(\g^*,+)$, i.e. a deformation of the Poisson algebra $\widehat{O}(G^*)$ whose Poisson structure comes from $\g$.
Edit: To extends on the classical case: for the quantization of $\g^*$, it is $S(\g^*)[[\hbar]]$ as an algebra because $\g^*$ is commutative, ie the trvial deformation of poylynomial function on $\g$, and the coproduct should be dual to the bracket of $\g$ at the first order somehow. Indeed it is given by the Baker-Campbell-Hausdorff formula: for $f\in \g^*$, $x,y\in \g$, $$\Delta(f)(x\otimes y)=f(BCH(x,y))$$
Taking again the subalgebra generated by $\hbar\g^*$, you get a flat deformation of $\widehat{S}(\g^*)=\widehat{O(G)}$ and the Hopf pairing is obtained by extending the pairing between $\g$ and $\g^*$. This is a standard fact that it agree with the pairing of Nicola's comment through the BCH formula. Indeed the standard coproduct on $O(G)$ is dual to the product of $G$, which can be expressed through the BCH formula. Hence the above coproduct is nothing but the pull back of the standard one on $O(G)$ through the iso $\widehat{S}(\g^*)\cong \widehat{O(G)}$.
It's worth mentionning that everything can be made fairly explicit in the semi-simple case, I guess it's done in Chari-Pressley. Kassel-Turaev's paper on biquantization of Lie bialgebra is also relevant here.
A: These questions are dicussed in 
N. Yu. Reshetikhin, L. A. Takhtadzhyan, L. D. Faddeev, “Quantization of Lie groups and Lie algebras”, (Full text in Russian is available from that page).
Let me follow Russian text in numeration of section and theorems.
You should look at section 2,  theorem 12, page 200.
It says that algebra U_q(h) is isomorphic to (mild extentsion) of Fun_q(G).
The isomorphism is explicitly given on generators for each of classical series A_n,B_n,C_n,D_n. (Exceptional groups seems  not mentioned).
Concerning the proof - the authors write "by direct calculation"...
The previous theorems in section 2, gives R-matrix description of the Fun_q(G)^*.

Informal remarks. This fact always seemed and seems now to me, as something surprising and un-understandable, because comultiplication on Fun_q(G) does not depend on "q" - it is standard matrix co-multiplication $\Delta(t_{ij}) = \sum_k t_{ik}\otimes t_{kj} $.
By duality the commultiplication should go to multiplication on U_q(g). 
Multiplication  depends on "q" and looks quite different from the simple formula above $\Delta(t_{ij}) = \sum_k t_{ik}\otimes t_{kj} $. So it is strange how one can match the two. Even for q=1, it seems quite surprising.
