Quantum traces and embeddings of stated skein algebras into quantum tori

Abstract

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