Skein Algebras and Quantum Groups

The $sl_n$-skein algebra of a surface provides a quantization of the $SL_n(\mathbb{C})$ character variety. For surfaces with boundary, this framework extends naturally to the stated skein algebra. We demonstrate how various aspects of quantum groups admit simple and transparent geometric interpretations through the lens of stated skein algebras. In particular, we show how the Schapiro–Shrader embedding of the quantized enveloping algebra into a quantum torus algebra arises from the quantum trace map. Time permitting, we will also present a geometric realization of the dual canonical basis of $\mathcal{O}_q(\mathfrak{sl}_3)$ using skeins.