Quantum Teichm\"uller spaces and quantum trace map
Thang T. Q. L\^e

TL;DR
This paper connects the quantum trace map and quantum Teichmüller space through skein algebras, providing a natural construction and concrete realization of these structures in quantum topology.
Contribution
It introduces a natural construction of the quantum trace map using the skein algebra and realizes the quantum Teichmüller space as a subalgebra of the skein algebra's skew field.
Findings
Quantum trace map constructed via skein algebra
Quantum Teichmüller space realized as a subalgebra
Provides a concrete algebraic framework for quantum topology
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 Teichm\"uller 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.
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