Quantum traces for representations of surface groups in SL 2 (ℂ)

We consider two different quantizations of the character variety consisting of all representations of surface groups in SL_2. One is the skein algebra considered by Przytycki-Sikora and Turaev. The other is the quantum Teichmuller space introduced by Chekhov-Fock and Kashaev. We construct a homomorphism from the skein algebra to the quantum Teichmuller space which, when restricted the classical case, corresponds to the equivalence between these two algebras through trace functions.