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 groups ${\overline{U}}_q^H(\mathfrak{g})$ 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 Müger 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.

Added Remark 4.9 and other minor corrections

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