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

Abstract

We study finite-dimensional representations of the Kauffman bracket skein algebra of a surface S. In particular, we construct invariants of such irreducible representations when the underlying parameter (Formula presented.) 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 (Formula presented.) to (Formula presented.) , 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.