On connection between associators and 6j-symbols

Abstract

We propose a new formula relating the rationalized Drinfeld associators obtained from the Knizhnik--Zamolodchikov equation to the quantum $6j$-symbols of the algebra $\mathcal{U}_q(\mathfrak{g})$: $ U_{q} = U \cdot Φ^{KZ}_{\mathbb{Q}} $. We prove this relation in the case of the $\mathfrak{sl}_2$ Lie algebra for the tensor cube of the fundamental and the first symmetric representation. We also check it perturbatively for the $\mathfrak{sl}_N$ algebra for representations $[2]\otimes [1] \otimes [1]$, $[2]\otimes [2] \otimes [1]$, the non-trivial cases with multiplicities --- $[2,1]^{\otimes 3}$, $[2,2]^{\otimes 3}$, and in the case of the $\mathfrak{so}_5$ Lie algebra for the tensor cube of the fundamental representation.