Triangular decomposition of skein algebras

By introducing a finer version of the Kauffman bracket skein algebra, 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. The new skein algebra of an ideal triangle has a simple presentation. This gives an easy proof of the existence of the quantum trace map of Bonahon and Wong. We also explain the relation between our skein algebra and the one defined by Muller, and use it to show that the quantum trace map can be extended to the Muller skein algebra.