2025/09/30 by Huang, Min, Wang, Zhihao
#FOS: Mathematics #Quantum Algebra (math.QA)
paper · doi:10.48550/arxiv.2509.25938
We introduce a quantum cluster algebra structure \mathscr Aω(\mathfrakS) inside the skew-field fractions \rm Frac(\widetilde\mathscrSω(\mathfrakS)) of the projected stated \rm SLn-skein algebra \widetilde\mathscrSω(\mathfrakS) (the quotient of the reduced stated \rm SLn-skein algebra by the kernel of the quantum trace map) for any triangulable pb surface \mathfrakS without interior punctures. To study the relationships among the projected \rm SLn-skein algebra \widetilde\mathscrSω(\mathfrakS), the quantum cluster algebra \mathscr Aω(\mathfrakS), and its quantum upper cluster algebra \mathscr Uω(\mathfrakS), we construct a splitting homomorphism for \mathscr Uω(\mathfrakS) and show that it is compatible with the splitting homomorphism for \widetilde\mathscrSω(\mathfrakS). When every connected component of \mathfrakS contains at least two punctures, this compatibility allows us to prove that \widetilde\mathscrSω(\mathfrakS) embeds into \mathscr Aω(\mathfrakS) by showing that the stated arcs joining two distinct boundary components of \mathfrakS (which generate \widetilde\mathscrSω(\mathfrakS)) 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 Uω(\mathfrakS), which, if true, would imply the equality \widetilde\mathscrSω(\mathfrakS)=\mathscr Aω(\mathfrakS)=\mathscr Uω(\mathfrakS).