Categories of weight modules for unrolled restricted quantum groups at roots of unity

Abstract

Motivated by connections to the singlet vertex operator algebra in the $\mathfrak{g}=\mathfrak{sl}_2$ case, we study the unrolled restricted quantum group $\overline{U}_q^{ H}(\mathfrak{g})$ for any finite dimensional complex simple Lie algebra $\mathfrak{g}$ at arbitrary roots of unity with a focus on its category of weight modules. We show that the braid group action naturally extends to the unrolled quantum groups and that the category of weight modules is a generically semi-simple ribbon category (previously known only for odd roots) with trivial Müger center and self-dual projective modules.