Quantum traces for $SL_n$-skein algebras

Abstract

We establish the existence of several quantum trace maps. The simplest one is an algebra map between two quantizations of the algebra of regular functions on the $SL_n$-character variety of a surface $\mathfrak{S}$ equipped with an ideal triangulation $\lambda$. The first is the (stated) $SL_n$-skein algebra $\mathscr{S}(\mathfrak{S})$. The second $\overline{\mathcal{X}}(\mathfrak{S},\lambda)$ is the Fock and Goncharov's quantization of their $X$-moduli space. The quantum trace is an algebra homomorphism $\bar{tr}^X:\overline{\mathscr{S}}(\mathfrak{S})\to\overline{\mathcal{X}}(\mathfrak{S},\lambda)$ where the reduced skein algebra $\overline{\mathscr{S}}(\mathfrak{S})$ is a quotient of $\mathscr{S}(\mathfrak{S})$. When the quantum parameter is 1, the quantum trace $\bar{tr}^X$ coincides with the classical Fock-Goncharov homomorphism. This is a generalization of the Bonahon-Wong quantum trace map for the case $n=2$. We then define the extended Fock-Goncharov algebra $\mathcal{X}(\mathfrak{S},\lambda)$ and show that $\bar{tr}^X$ can be lifted to $tr^X:\mathscr{S}(\mathfrak{S})\to\mathcal{X}(\mathfrak{S},\lambda)$. We show that both $\bar{tr}^X$ and $tr^X$ are natural with respect to the change of triangulations. When each connected component of $\mathfrak{S}$ has non-empty boundary and no interior ideal point, we define a quantization of the Fock-Goncharov $A$-moduli space $\overline{\mathcal{A}}(\mathfrak{S},\lambda)$ and its extension $\mathcal{A}(\mathfrak{S},\lambda)$. We then show that there exist quantum traces $\bar{tr}^A:\overline{\mathscr{S}}(\mathfrak{S})\to\overline{\mathcal{A}}(\mathfrak{S},\lambda)$ and $tr^A:\mathscr{S}(\mathfrak{S})\hookrightarrow\mathcal{A}(\mathfrak{S},\lambda)$, where the second map is injective, while the first is injective at least when $\mathfrak{S}$ is a polygon. They are equivalent to the $X$-versions but have better algebraic properties.