Triangular decomposition of skein algebras and quantum Teichmüller spaces

We show how to decompose the Kauffman bracket skein algebra of a surface into elementary blocks corresponding to the triangles in an ideal triangulation of the surface. This gives an easy proof of the existence of the quantum trace map of Bonahon and Wong. We also explain the relation between the Kauffman bracket skein algebra and the quantum Teichmüller space.
The author is partially supported by the NSF.