# The quantum trace as a quantum non-abelianization map

## TL;DR

This paper establishes a deep connection between the quantum trace, skein algebras, and character varieties, revealing a non-abelianization map that generalizes and relates to existing structures in quantum topology and geometry.

## Contribution

It introduces a novel algebraic non-abelianization map that links skein algebras with character varieties, extending previous classifications and interpretations.

## Key findings

- Classification of irreducible representations at roots of unity
- Re-interpretation of the quantum trace as a non-commutative deformation
- Induction of a birational morphism for closed surfaces

## Abstract

We prove that the balanced Chekhov-Fock algebra of a punctured triangulated surface is isomorphic to a skein algebra which is a deformation of the algebra of regular functions of some abelian character variety. We first deduce from this observation a classification of the irreducible representations of the balanced Chekhov-Fock algebra at odd roots of unity, which generalizes to open surfaces the classification of Bonahon, Liu and Wong. We re-interpret Bonahon and Wong's quantum trace map as a non-commutative deformation of some regular morphism between this abelian character variety and the SL2-character variety. This algebraic morphism shares many resemblance with the non-abelianization map of Gaiotto, Moore, Hollands and Neitzke. When the punctured surface is closed, we prove that this algebraic non-abelianization map induces a birational morphism between a smooth torus and the relative SL2 character variety.