As is well-known, one can use the coquasi-triangular structure $\cal R$ of $U_q(\frak{g})$ to give it's category of (right) modules $\cal{M}_{U_q(\frak{g}})$ the structure of a braided monoidal category. Moreover, one use $\cal R$ to give $\cal{M}_{U_q(\frak{g}})$ the structure of a ribbon category.

On the coordinate algebra side $G_q$, one can also the coquasi-triangular structure $r$ to give $\cal{M}^{G_q}$ the structure of a braided monoidal category. What I would like to know is can one use $r$ to give $\cal{M}^{G_q}$ the structure of a ribbon category.

As is well-known, one can use the coquasi-triangular structure $\cal R$ of $U_q(\frak{g})$ to give it's category of (right) modules $\cal{M}_{U_q(\frak{g})}$ the structure of a braided monoidal category. Moreover, one use $\cal R$ to give $\cal{M}_{U_q(\frak{g})}$ the structure of a ribbon category.

On the coordinate algebra side $G_q$, one can also the coquasi-triangular structure $r$ to give $\cal{M}^{G_q}$ the structure of a braided monoidal category. What I would like to know is can one use $r$ to give $\cal{M}^{G_q}$ the structure of a ribbon category.