  We review the representation theory of the quantum group $U_\epsilon
sl_2\mathbb{C}$ at a root of unity $\epsilon$ of odd order, focusing on
geometric aspects related to the 3-dimensional quantum hyperbolic field
theories (QHFT). Our analysis relies on the quantum coadjoint action of De
Concini-Kac-Procesi, and the theory of Heisenberg doubles of Poisson-Lie groups
and Hopf algebras. We identify the 6j-symbols of generic representations of
$U_\epsilon sl2\mathbb{C}$, the main ingredients of QHFT, with a bundle
morphism defined over a finite cover of the algebraic quotient
$PSL_2\mathbb{C}/!/PSL_2\mathbb{C}$, of degree two times the order of
$\epsilon$. It is characterized by a non Abelian 3-cocycloid identity deforming
the fundamental five term relation satisfied by the classical dilogarithm
functions, that relates the volume of hyperbolic 3-polyhedra under
retriangulation, and more generally, the simplicial formulas of Chern-Simons
invariants of 3-manifolds with flat $sl_2\mathbb{C}$-connections.
