  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.
