Motivated by [this question](https://mathoverflow.net/questions/25423/the-inverse-of-a-universal-r-matrix-for-quantized-universal-enveloping-algebra-of), I'm going to ask a much more direct one: Let $R$ be a universal R-matrix 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  

$$
\langle R,u^i\_j \otimes u^r\_s\rangle = q^{\delta\_{ir}}\delta\_{ij}\delta\_{rs} + (q-q^{-1})\theta (i-r)\delta\_{is}\delta\_{jr},
$$
where $\theta$ denotes the Heaviside symbol. 



Formula for the Matrix Elements of the Inverse of special linear Universal R-Matrix of Uq(sln)
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


$\langle R^{-1},u^i\_j\otimes u^r\_s \rangle = q^{-1}(q^{-\delta\_{ir}} \delta\_{ij}\delta\_{rs} + (q^{-1}-q)\theta(i-r)\delta(is)\delta(jr))$.


As I explained, the formula you wrote there is really for $q^{\frac 12} \langle R,u^i\_j\otimes u^r\_s \rangle$.


