Quantum traces and embeddings of stated skein algebras into quantum tori

The stated skein algebra of a punctured bordered surface (or equivalently, marked surface) is generalization the well-known Kauffman bracket unmarked surfaces and can be considered as an extension quantum special linear group \({\mathcal {O}}_{q^2}(SL_2)\) from bigon to general surfaces. We show that with non-empty boundary embedded into tori in two different ways. first embedding quantization map expressing trace closed curve terms shear coordinates enhanced Teichmüller space, lift Bonahon-Wong’s map. second lambda length decorated Muller’s explain relation between maps. also cluster Muller equal reduced version algebra. As applications we orderly finitely generated Noetherian domain calculate its Gelfand-Kirillov dimension.