This question is related to this one (and indeed the goals are similar).

Let $N$ be odd and consider the braided fusion category $\mathcal{C}$ (actually modular) obtained from $U_q\mathfrak{so}_N$ at $q=e^{\pi i/(2N)}$ in the usual way (i.e. take the quotient of the category of tilting modules of Lusztig's integral form by the tensor ideal of negligible morphisms). This is sometimes denoted $SO(N)_2$ by physicists. Let $V$ denote the object in $\mathcal{C}$ analogous to the fundamental *spin* representation. Then the braid group $\mathcal{B}_n$ acts on the simple $End(V^{\otimes n})$-modules $Hom(W,V^{\otimes n})$ irreducibly, since in this case the image of $\mathcal{B}_n$ generates the centralizer algebras.

**Question:** how can one explicitly describe the braid group representations, say up to $n=5$? I.e. is there a uniform way of writing down the matrices of the generators of $\mathcal{B}_n$?

What I know:

- For $q$ generic one can just use the $R$-matrix.
- For $N=3$ this is a (quotient of) the Temperley-Lieb algebras
- For $N=5$ this is a (quotient of) BMW-algebras (since $\mathfrak{so}_5$ is $\mathfrak{sp}_4$).
- For $N=7$ Westbury has a description MR2388243.
- For $N=k^2$ the category is integral and in fact group-theoretical

So for $N\leq 7$ or a perfect square, one can get a useful description of the braid group action using the irreps of the quotient algebras or by the Drinfeld double construction. It is too much to ask for a description as a quotient of $\mathcal{B}_n$ for $q$ generic, but perhaps there is a way for this particular value of $q$.