Quantum duality maps, skein algebras and their ensemble compatibility

We generalize the quantum duality map ${\mathbb{I}}_{\mathcal{A}}$ of Allegretti–Kim [AK17] for punctured closed surfaces to general marked surfaces. When the marked surface has no interior marked points, we investigate its compatibility with the quantum duality map ${\mathbb{I}}_{\mathcal{X}}$ on the dual side based on the quantum bracelets basis [Thu14, MQ23]. Our construction factors through reduced stated skein algebras, based on the quantum trace maps [Lê18] together with an appropriate way of \emph{skein lifting} of integral $\mathcal{A}$-laminations. We also give skein theoretic proofs for some expected properties of Laurent expressions, and positivity of structure constants for marked disks and a marked annulus.