Résumé
Interactions between Hyperbolic Geometry, Quantum Topology and Number Theory, W. Neumann et al. (Ed.) (2011) Volume 541 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.