Question

Motivated by this question, 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.

Answer 1

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 $q^{\frac 12} \langle R,u^i_j\otimes u^r_s \rangle$.