Quantum trace maps for skein algebras
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Yu, Tao
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Abstract
We study quantizations of $SL_n$-character varieties, which appears as moduli spaces for many geometric structures. Our main goal is to establish the existence of several quantum trace maps.
Given a surface $\mathfrak{S}$ and an ideal triangulation $\lambda$, we define an algebra homomorphism $\overline{tr}^X_\lambda: \overline{\mathscr{S}}(\mathfrak{S}) \to \overline{\mathcal{X}}(\mathfrak{S},\lambda)$ from the reduced skein algebra $\overline{\mathscr{S}}(\mathfrak{S})$, which is a quotient of the stated $SL_n$-skein algebra $\mathscr{S}(\mathfrak{S})$, to the Fock-Goncharov algebra $\overline{\mathcal{X}}(\mathfrak{S},\lambda)$. When the quantum parameter is 1, $\overline{\mathscr{S}}(\mathfrak{S})$ and $\overline{\mathcal{X}}(\mathfrak{S},\lambda)$ reduce to the algebras of regular functions on the character variety and the $X$-coordinate chart of Fock-Goncharov respectively, and $\overline{tr}^X_\lambda$ gives the coordinate expressions of the classical trace functions. This is a generalization of the famous 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 $\overline{tr}^X_\lambda$ can be lifted to an extended quantum trace $tr^X_\lambda: \mathscr{S}(\mathfrak{S}) \to \mathcal{X}(\mathfrak{S}, \lambda)$.
When each connected component of $\mathfrak{S}$ has non-empty boundary and no interior ideal point, we define a quantization $\overline{\mathcal{A}}(\mathfrak{S}, \lambda)$ of the Fock-Goncharov $A$-moduli space associated to $SL_n$ and its extension $\mathcal{A}(\mathfrak{S}, \lambda)$. We then show that there exist the $A$-versions of the quantum traces $\overline{tr}^A: \overline{\mathscr{S}}(\mathfrak{S}) \to \overline{\mathcal{A}}(\mathfrak{S}, \lambda)$ and $tr^A: \mathscr{S}(\mathfrak{S}) \hookrightarrow \mathcal{A}(\mathfrak{S}, \lambda)$ with simple transitions to the $X$-versions. We show that the $A$-versions have better algebraic properties and use them to construct coordinate change isomorphisms that relate the quantum traces in different triangulations.
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2023-04-20
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Dissertation