Consider a Unitary Modular Tensor Category constructed from the quantum group of some compact, simple and simply-connected Lie group $U\_q(G)$ at $q=e^{2\pi i/(k+h)}$ for some integer $k$. In general, the associator ("$F$-symbols", i.e., the pentagon equations) is very hard to compute explicitly. To the best of my knowledge, only $SU(2)$ is completely known (the quantum $6j$ symbols for arbitrary $k$, cf. Kirillov-Reshetikhin), and a few other specific cases (e.g., $SU(3)$ for some small values of $k$, and the Ising category $Spin(2n+1)$ at $k=1$).


I am looking for some other cases where the $F$-symbols have been worked out explicitly. I am particularly interested in the $B\_r,C\_r,D\_r$ series at some non-trivial value of $k$.



F-symbols for compact Lie groups
We do The $B$ series at level 2 in our paper [classifying metaplectic fusion categories](https://arxiv.org/abs/1608.03762).


