An embedding of skein algebras of surfaces into localized quantum tori from Dehn–Thurston coordinates

Detcherry, Renaud¹ ; Santharoubane, Ramanujan²

¹ Institut de Mathématiques de Bourgogne, UMR 5584 CNRS, Université de Bourgogne, Dijon, France
² Laboratoire Mathématiques d’Orsay, UMR 8628 CNRS, Université Paris-Saclay, Orsay, France

Geometry & topology, Tome 29 (2025) no. 1, pp. 313-348.

Résumé

We construct embeddings of Kauffman bracket skein algebras of surfaces (either closed or with boundary) into localized quantum tori using the action of the skein algebra on the skein module of the handlebody. We use those embeddings to study representations of Kauffman skein algebras at roots of unity and get a new proof of Bonahon and Wong’s unicity conjecture. Our method allows one to explicitly reconstruct the unique representation with fixed classical shadow, as long as the classical shadow is irreducible with image not conjugate to the quaternion group.

Détail

DOI : 10.2140/gt.2025.29.313

Keywords: skein algebras, quantum tori, Azumaya locus

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