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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.

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    $\begingroup$ What have you tried? If you tell us this then we will be better able to help you. And it helps us feel that we are not just doing your homework for you. $\endgroup$ – user1729 Jun 5 '14 at 8:45
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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.

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