Résumé
We review the representation theory of the quantum group U(epsilon)sl(2)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)sl(2)C, the main ingredients of QHFT, with a bundle morphism defined over a finite cover of the algebraic quotient PSL2C//PSL2C, of degree two times the order of epsilon. It is characterized by a non Abelian 3-cocycloid identity deforming the fundamental live 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)C-connections.