Faithfullness of geometric action of skein algebras

TL;DR

This paper proves that the geometric action of skein algebras on handlebody skein modules is faithful precisely when the quantum parameter is not a root of unity, clarifying conditions for faithfulness.

Contribution

It establishes a necessary and sufficient condition for the faithfulness of skein algebra actions based on the quantum parameter's properties.

Findings

Faithful action occurs iff the quantum parameter is not a root of 1.

Provides a characterization of when skein algebra actions are faithful.

Clarifies the role of quantum parameters in skein algebra representations.

Abstract

We show that the action of the Kauffman bracket skein algebra of a surface $\Sigma$ on the skein module of the handlebody bounded by $\Sigma$ is faithful if and only if the quantum parameter is not a root of 1.