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

TL;DR

This paper explores the structure of weight modules for unrolled restricted quantum groups at roots of unity, revealing their semi-simple ribbon category structure and connections to vertex operator algebras.

Contribution

It extends the braid group action to unrolled quantum groups and characterizes the category of weight modules as a semi-simple ribbon category with trivial M"{u}ger center.

Findings

Braid group action extends to unrolled quantum groups.

Category of weight modules is a semi-simple ribbon category.

Projective covers of simple modules are self-dual.

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 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.