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

Matthew Rupert

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.

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 case, we study the unrolled restricted quantum groups 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 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.

Paper Structure

This paper contains 8 sections, 18 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Results
  3. Preliminaries
  4. The Unrolled Restricted Quantum Group $\overline{U}_q^H(\mathfrak{g})$
  5. Braid Group Action on $U_{q}^H(\mathfrak{g})$
  6. Representation Theory of $\overline{U}_q^{H}(\mathfrak{g})$
  7. Categorical Structure
  8. Duality

Key Result

Proposition 1.1

The action of the braid group $\mathcal{B}_{\mathfrak{g}}$ on $U_q(\mathfrak{g})$ can be extended naturally to the unrolled quantum group $U_q^H(\mathfrak{g})$.

Theorems & Definitions (33)

  • Proposition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Definition 2.1
  • Definition 2.2
  • Definition 3.1
  • Remark 3.2
  • Definition 3.3
  • Proposition 3.4
  • ...and 23 more