Quantum cluster algebra realization of stated SLn-skein algebras and rotation-invariant bases for polygons

We construct a quantum cluster structure on the skew-field of fractions of the stated SL_n-skein algebra for a triangulable pb surface without interior punctures. We prove that the stated SL_n-skein algebra is a subalgebra of the corresponding quantum cluster algebra. For the polygon, we prove that the stated SL_n-skein algebra coincides with both the quantum cluster algebra and the quantum upper cluster algebra, after localization at the frozen variables. Moreover, we show that the theta basis of the quantum cluster algebra yields a rotation-invariant basis of the skein algebra with several desirable properties, including positivity and a natural parametrization. In the special case where n=3, for the bigon, we further provide a complete web-theoretic characterization of the cluster variables and clusters. Consequently, we establish a web interpretation of the dual canonical basis of O_q(SL_3). This is a joint work with Peigen Cao and Zhihao Wang.