The quantum trace as a quantum non-abelianization map

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