QUANTUM TEICHMÜLLER SPACES AND QUANTUM TRACE MAP

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.

THE QUANTUM TEICHMÜLLER SPACE AS A NONCOMMUTATIVE ALGEBRAIC OBJECT

We consider the quantum Teichmüller space of the punctured surface introduced by Chekhov–Fock–Kashaev, and formalize it as a noncommutative deformation of the space of algebraic functions on the Teichmüller space of the surface. In order to apply it in 3-dimensional topology, we put more attention to the details involving small surfaces.