Quantum traces and embeddings of stated skein algebras into quantum tori

TL;DR

This paper introduces two quantum trace embeddings of the stated skein algebra of punctured bordered surfaces into quantum tori, extending previous trace maps and relating to quantum cluster algebras, with applications in algebraic properties.

Contribution

It constructs two distinct embeddings of the stated skein algebra into quantum tori, extending known quantum trace maps and connecting to quantum cluster algebra theory.

Findings

Embedded skein algebra into quantum tori in two ways

Extended quantum trace maps for different Teichmüller space coordinates

Proved skein algebra is finitely generated, Noetherian, and calculated its Gelfand-Kirillov dimension

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 Teichm\"uller space, and is a lift of Bonahon-Wong's quantum trace map. The second embedding can be considered as a quantization of the map expresses the trace of a closed curve in terms of the lambda length coordinates of the decorated Teichm\"uller 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.