What is a Skein Module?

A skein module is a module associated to a three-dimensional manifold by considering linear combinations of links in the manifold, modulo properly chosen skein relations.

Roots of Unity

When $q$ is generic, the center of $S_q^{SL_2}(\Sigma)$ is trivial. When $q$ is a root of unity, the center becomes much more rich. This idea of quantum groups at roots of unity is a common theme.

The quantum group $\mathcal{O}_q(SL_3)$

$\mathcal{O}_q(SL_3)$ is the quantized coordinate ring of $SL_3$, and should be thought of as the dual construction of the much more famous $\mathcal{U}_q(\mathfrak{sl}_3)$.