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

TL;DR

This paper investigates finite-dimensional representations of the Kauffman skein algebra of a surface, introducing invariants at roots of unity and revealing miraculous cancellations in the quantum trace homomorphism.

Contribution

It constructs new invariants for irreducible representations of the skein algebra at roots of unity, based on miraculous cancellations in the quantum trace homomorphism.

Findings

Invariant points in the character variety are associated with irreducible representations.

Miraculous cancellations are crucial for constructing these invariants.

The work lays groundwork for future representations of the skein algebra.

Abstract

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.