Quantum traces for representations of surface groups in SL_2

TL;DR

This paper explores two different quantum approaches to the character variety of surface group representations in SL_2, establishing a homomorphism linking skein algebra and quantum Teichmuller space.

Contribution

It constructs a homomorphism connecting skein algebra and quantum Teichmuller space, unifying two quantizations of surface group representations.

Findings

Established a homomorphism between skein algebra and quantum Teichmuller space.

Demonstrated the classical limit correspondence between the two quantum approaches.

Unified different quantizations of surface group representations in SL_2.

Abstract

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.