We extend some results of Bonahon, Bullock, Turaev and Wong concerning the skein algebras of closed surfaces to Lê's stated skein algebra associated to open surfaces. We prove that the stated skein algebra with deforming parameter +1 embeds canonically into the centers of the stated skein algebras whose deforming parameter is an odd root unity. We also construct an isomorphism between the stated skein algebra at +1 and the algebra of regular function of a generalization of the SL2-character variety of the surface. As a result, we associate to each isomorphism class of irreducible or local representations of the stated skein algebra, an invariant which is a point in the character variety.
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 SL 2 -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 SL 2 character variety.
We show that the double cobar construction, Ω 2 C * (X), of a simplicial set X is a homotopy BValgebra if X is a double suspension, or if X is 2-reduced and the coefficient ring contains the field of rational numbers Q. Indeed, the Connes-Moscovici operator defines the desired homotopy BV-algebra structure on Ω 2 C * (X) when the antipode S : ΩC * (X) → ΩC * (X) is involutive. We proceed by defining a family of obstructions O n : C * (X) → C * (X) ⊗n , n ≥ 2 by computing S 2 − Id. When X is a suspension, the only obstruction remaining is O 2 := E 1,1 − τE 1,1 where E 1,1 is the dual of the ⌣ 1 -product. When X is a double suspension the obstructions vanish.
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