Dilogarithme Quantique et 6j-Symboles Cycliques

Let ${\mathcal{W}}_N$ be a quantized Borel subalgebra of $U_q(sl(2,\text{mc}))$, specialized at a primitive root of unity $\omega =\exp (2i\pi \mathrm{/}N)$ of odd order $N>1$. One shows that the $6j$-symbols of cyclic representations of ${\mathcal{W}}_N$ are representations of the canonical element of a certain extension of the Heisenberg double of ${\mathcal{W}}_N$. This canonical element is a twisted $q$-dilogarithm. In particular, one gives explicit formulas for these $6j$-symbols, and one constructs partial symmetrizations of them, the c-$6j$-symboles. The latters are at the basis of the construction of the quantum hyperbolic invariants of 3-manifolds.