# Dual Pairing between $U_q(\mathfrak{su}(n))$ and $\mathcal{O}(SU_q(n))$

In page 327 of Klymik and Schmudgen`s book Quantum Groups and their representations, theorem 18 reads:

There exists unique dual pairings of the pairs:

• $U_q(gl_n)$ and $\mathcal{O}(GL_q(n))$,
• SL_q(n)...
• O_q(2n)...
• O_q(2n + 1)... etc...

Where $\mathcal{O}(G_q)$ denotes the quantized algebra of coordinate functions (defined and studied in the book) and $U_q(g)$ is the quantized universal enveloping algebra of the Lie algebra $g$.

But there is no mention of $\mathcal{O}(SU_q(n))$, nor $\mathcal{O}(U_q(n))$. My question is, is there a similar dual pairing of Hopf algebras for $U_q(su(n))$ and $\mathcal{O}(SU_q(n))$ (respectively, $U_q(U(n))...$)? If so, is it unique?

Bonus questions: What about noncompact quantum groups, particularly $U_q(su(n, m))$ and $\mathcal{O}(SU_q(n, m))$?