4
$\begingroup$

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.

Improve this question
$\endgroup$
1
  • 1
    $\begingroup$ $U_q$ and $G_q$ are dual Hopf algebras, so up to some minor issues (f.d.) modules over the one is the same as comodules over the other. So these categories are really the same, as are $r$ and $R$. So yes, you can definitely do the same with the ribbon structure. In fact working on the $G_q$ side is somewhat more convenient because you don't need any kind of completion for these elements to be well defined. $\endgroup$
    – Adrien
    Oct 19, 2013 at 16:07

0

Reset to default

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.