Categories of Weight Modules for Unrolled Restricted Quantum Groups at Roots of Unity

Motivated by connections to the singlet vertex operator algebra in the $\mathfrak{g}={\mathfrak{s}\mathfrak{l}}_2$ case, we study the unrolled restricted quantum groups ${\bar{U}}_q^H(\mathfrak{g})$ at arbitrary roots of unity with a focus on its category of weight modules. We show that the braid group action on the Drinfeld-Jimbo algebra $U_q(\mathfrak{g})$ 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 M"{u}ger center. Projective covers of simple modules are shown to be self-dual, and some preliminary connections to the higher rank singlet vertex operator algebras are motivated.