Motivated by this question, I'm going to ask a much more direct one: Let $R$ be a universal Rmatrix for the quasitriangular Hopf algebra ${\cal U}_q ({\mathfrak sl}_N)$, $~$ $R ^{1}$ its inverse, and $\langle \cdot,\cdot \rangle$ the dual pairing between ${\cal U}_q ({\mathfrak sl}_N)$ and $SU_q[N]$, do we have a formula for
$$
\langle R^{1},u^i_j \otimes u^r_s\rangle
$$
analogous to the formula
$$
\langle R,u^i_j \otimes u^r_s\rangle = q^{\delta_{ir}}\delta_{ij}\delta_{rs} + (qq^{1})\theta (ir)\delta_{is}\delta_{jr},
$$
where $\theta$ denotes the Heaviside symbol.



John, see my answer there. It is as the OP thought. The formula for $R^{1}$ on the vector represenation can only be the inverse of the formula for $R$, namely $\langle R^{1},u^i_j\otimes u^r_s \rangle = q^{1}(q^{\delta_{ir}} \delta_{ij}\delta_{rs} + (q^{1}q)\theta(ir)\delta(is)\delta(jr))$. As I explained, the formula you wrote there is really for $q^{\frac 12} \langle R,u^i_j\otimes u^r_s \rangle$. 

