Uprolling Unrolled Quantum Groups

TL;DR

This paper constructs and analyzes categories of modules over unrolled quantum groups at roots of unity, exploring their properties and potential connections to vertex operator algebras, especially in higher rank cases.

Contribution

It introduces new families of commutative algebra objects in the category of weight modules for unrolled quantum groups and studies their module categories, including conditions for finiteness and non-degeneracy.

Findings

Categories of local modules are characterized for various algebra objects.

Conditions for categories to be finite, non-degenerate, and ribbon are derived.

Examples suggest connections to higher rank triplet vertex algebras.

Abstract

We construct families of commutative (super) algebra objects in the category of weight modules for the unrolled restricted quantum group $\overline{U}_q^H(\mfg)$ of a simple Lie algebra $\mfg$ at roots of unity, and study their categories of local modules. We determine their simple modules and derive conditions for these categories being finite, non-degenerate, and ribbon. Motivated by numerous examples in the $\mfg=\mathfrak{sl}_2$ case, we expect some of these categories to compare nicely to categories of modules for vertex operator algebras. We focus in particular on examples expected to correspond to the higher rank triplet vertex algebra $W_Q(r)$ of Feigin and Tipunin \cite{FT} and the $B_Q(r)$ algebras of \cite{C1}.