Representations of the Kauffman bracket skein algebra I: invariants and miraculous cancellations

We study finite-dimensional representations of the Kauffman skein algebra of a surface S. In particular, we construct invariants of such irreducible representations when the underlying parameter q is a root of unity. The main one of these invariants is a point in the character variety consisting of group homomorphisms from the fundamental group of S to SL_2(C), or in a twisted version of this character variety. The proof relies on certain miraculous cancellations that occur for the quantum trace homomorphism constructed by the authors. These miraculous cancellations also play a fundamental role in subsequent work of the authors, where novel examples of representations of the skein algebra are constructed.Comment: Version 3: Improvements in the exposition following referee reports. This version also fixes a small gap in the proof of the miraculous cancellations of Theorems 4 and 21, originally caused by an incorrect interpretation of the reference [Bu] used to create a shortcut in the computations; the results are unchanged, and the modifications to the proof very minima