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.
Categories: math.GT