Georgia Institute of Technology College of Sciences

This talk provides an elementary introduction to skein algebras of surfaces, which serve as quantizations of $SL_n$-character varieties. For surfaces with boundary, we extend this framework to stated skein algebras, demonstrating how they provide simple and transparent geometric interpretations of various quantum group structures.

Specifically, we present a geometric realization of the dual canonical basis of $\mathscr{O}_q(\mathfrak{sl}_n)$ using skeins for $n=2$ and $n=3$. If time permits, we will also show how the skein algebra framework can be used to recover the Shapiro–Schrader embedding of the quantized enveloping algebra into a quantum torus algebra.