How to show that $\mathcal{O}_q[U]$ is isomorphic to $U_q(\mathfrak{n})$?

Question

Let $U$ be the positive unipotent radical of $SL_n$ and $\mathfrak{n}$ the Lie algebra of $U$. How to show that $\mathcal{O}_q[U]$ is isomorphic to $U_q(\mathfrak{n})$? Here $\mathcal{O}_q[U]$ is the quantum coordinate ring of $U$ and $U_q(\mathfrak{n})$ is the quantized universal enveloping algebra of $\mathfrak{n}$. Thank you very much.

Answer 1

If I remember correctly this can be found in Joseph "Quantum groups and primitive ideals", I'll later add the correct reference. EDIT It is the content of 9.2.12, if I am not mistaken (notations do not make this an easy book to jump in...)

One way of seeing it can be the following. It follows from the analogous statement about $U_q(\mathfrak b^+)\simeq {\cal O}(B_-)$. This can be seen as a consequence of the fact that there exists a perfect Hopf algebra pairing between $U_q(\mathfrak b^+)$ and $U_q(\mathfrak b^-)^{op}$ (explicit, e.g. Klimyk-Schmudgen page 184) using which one can identify $U_q(\mathfrak b^+)$ with ${\cal O}_q(B_-)$. There is a little bit of hand-waving going on here because one has to be very clear whether he is working with $\mathbb C$--algebras or $\mathbb C[[\hbar ]]$ algebras or $\mathbb C(q)$--algebras.

A possible way of seeing this is as an issue of Drinfeld's quantum duality principle, since w. r. to the natural Poisson--Lie group structure $\pi$, $(B^+,\pi)$ and $(B^-,-\pi)$ are Poisson dual groups.