Quantum Teichm\"uller spaces and quantum trace map

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.