Volume 15, issue 3 (2011)

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Quantum traces for representations of surface groups in $\mathrm{SL}_2(\mathbb{C})$

Francis Bonahon and Helen Wong

Geometry & Topology 15 (2011) 1569–1615
Abstract

We relate two different quantizations of the character variety consisting of all representations of surface groups in ${SL}_{2}$. One is the Kauffman skein algebra considered by Bullock, Frohman and Kania-Bartoszyńska, Przytycki and Sikora, and Turaev. The other is the quantum Teichmüller space introduced by Chekhov and Fock and by Kashaev. We construct a homomorphism from the skein algebra to the quantum Teichmüller space which, when restricted to the classical case, corresponds to the equivalence between these two algebras through trace functions.

Keywords
Kauffman skein relation, character variety, surface group, skein module, skein algebra, quantum Teichmüller theory
Mathematical Subject Classification 2010
Primary: 14D20, 57M25, 57R56