Quantum Teichmüller spaces and quantum trace map

Abstract

We show how the quantum trace map of Bonahon and Wong can be constructed in a natural way using the skein algebra of Muller, which is an extension of the Kauffman bracket skein algebra of surfaces. We also show that the quantum Teichmüller space of a marked surface, defined by Chekhov-Fock (and Kashaev) in an abstract way, can be realized as a concrete subalgebra of the skew field of the skein algebra.

Final version. To appear in Journal of the Institute of Mathematics of Jussieu

References in corpus (1)

• Character varieties, A-polynomials, and the AJ Conjecture

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• Quantum torus methods for Kauffman bracket skein modules
• Quantum duality maps, skein algebras and their ensemble compatibility
• The geometry of cluster varieties from surfaces
• An embedding of skein algebras of surfaces into quantum tori from Dehn-Thurston coordinates
• A comparison between $S{L}_n$ spider categories
• ${\mathrm{SL}}_3$-laminations as bases for ${\mathrm{PGL}}_3$ cluster varieties for surfaces
• Stated skein algebras and their representations
• Quantum traces for ${\mathrm{SL}}_n(\mathbb{C})$: The case $n=3$