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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
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 SL2. 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
References
Publication
Received: 10 January 2011
Revised: 10 January 2011
Accepted: 18 July 2011
Published: 19 September 2011
Proposed: Walter Neumann
Seconded: Jean-Pierre Otal, David Gabai
Authors
Francis Bonahon
Department of Mathematics
University of Southern California
3620 S Vermont Ave, KAP 108
Los Angeles CA 90089-2532
USA
http://www-rcf.usc.edu/~fbonahon/
Helen Wong
Department of Mathematics
Carleton College
1 North College St
Northfield MN 55057
USA
http://people.carleton.edu/~hwong