Quantum duality maps, skein algebras and their ensemble compatibility

TL;DR

This paper extends quantum duality maps to general marked surfaces, explores their compatibility, and provides skein-theoretic proofs for properties like Laurent positivity in specific cases.

Contribution

It generalizes the quantum duality map for punctured surfaces to all marked surfaces and establishes compatibility with dual maps using skein algebra techniques.

Findings

Compatibility of quantum duality maps with dual structures for marked surfaces.

Skein-theoretic proofs of Laurent positivity and structure constant positivity.

Construction of the quantum duality map via skein lifting and quantum trace maps.

Abstract

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 [L\^e18] 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.