Quantum cluster realization for projected stated ${\rm SL}_n$-skein algebras

We introduce a quantum cluster algebra structure $\mathscr A_\omega(\mathfrak{S})$ inside the skew-field fractions ${\rm Frac}\bigl(\widetilde{\mathscr{S}}_\omega(\mathfrak{S})\bigr)$ of the projected stated ${\rm SL}_n$-skein algebra $\widetilde{\mathscr{S}}_\omega(\mathfrak{S})$ (the quotient of the reduced stated ${\rm SL}_n$-skein algebra by the kernel of the quantum trace map) for any triangulable pb surface $\mathfrak{S}$ without interior punctures. To study the relationships among the projected ${\rm SL}_n$-skein algebra $\widetilde{\mathscr{S}}_\omega(\mathfrak{S})$, the quantum cluster algebra $\mathscr A_\omega(\mathfrak{S})$, and its quantum upper cluster algebra $\mathscr U_\omega(\mathfrak{S})$, we construct a splitting homomorphism for $\mathscr U_\omega(\mathfrak{S})$ and show that it is compatible with the splitting homomorphism for $\widetilde{\mathscr{S}}_\omega(\mathfrak{S})$. When every connected component of $\mathfrak{S}$ contains at least two punctures, this compatibility allows us to prove that $\widetilde{\mathscr{S}}_\omega(\mathfrak{S})$ embeds into $\mathscr A_\omega(\mathfrak{S})$ by showing that the stated arcs joining two distinct boundary components of $\mathfrak{S}$ (which generate $\widetilde{\mathscr{S}}_\omega(\mathfrak{S})$) are, up to multiplication by a Laurent monomial in the frozen variables, exchangeable cluster variables. We further conjecture that these exchangeable cluster variables generate the quantum upper cluster algebra $\mathscr U_\omega(\mathfrak{S})$, which, if true, would imply the equality $\widetilde{\mathscr{S}}_\omega(\mathfrak{S})=\mathscr A_\omega(\mathfrak{S})=\mathscr U_\omega(\mathfrak{S})$.