Categories of Weight Modules for Unrolled Quantum Groups and Connections to Vertex Operator Algebras

Abstract

Recently, numerous connections between the categories of modules $\mathrm{Rep}{\langle s \rangle} \mcM(r)$ for the singlet vertex operator algebra $\mcM(r)$ and $\mathrm{Rep}{wt}\overline{U}_q^H(\mathfrak{sl}_2)$ for the unrolled restricted quantum group $\overline{U}_q^H(\mathfrak{sl}2)$ at $2r$-th root of unity have been established. This has led to the conjecture that these categories are ribbon equivalent. In this thesis, we focus on extending the known connections between the singlet vertex algebra and unrolled quantum groups to the $B_r$ vertex algebra, and developing unrolled quantum groups in higher rank. In the first portion of this thesis, we use the conjectural ribbon equivalence between $\mathrm{Rep}{wt}\overline{U}q^H(\mathfrak{sl}2)$ and $\mathrm{Rep}{\langle s \rangle} \mcM(r)$ to identify algebra objects $\mcA_r$ associated to the $B_r$ vertex operator algebra and show that the properties of its category of local modules, $\mathrm{Rep}^0\mcA_r$, compare nicely to that of $\mathrm{Rep}{\langle s \rangle} B_r$. For the second portion of this thesis, we begin by showing that the category of weight modules $\mathrm{Rep}_{wt}\overline{U}_q^H(\mfg)$ for the unrolled restricted quantum group $\overline{U}q^H(\mfg)$ associated to a simple Lie algebra $\mfg$ is generically semisimple and ribbon with trivial M"{u}ger center. We then construct families of commutative (super) algebra objects in $\mathrm{Rep}{wt}\overline{U}_q^H(\mfg)$ and study their categories of local modules. Their irreducible modules are determined and conditions for these categories being finite, non-degenerate, and ribbon are derived. Among these commutative algebra objects are examples whose categories of local modules are expected to compare nicely to module categories of the higher rank triplet $W_Q(r)$ and $B_Q(r)$ vertex algebras. Lastly, we restrict to the case of $\overline{U}_i^H(\mathfrak{sl}_3)$. The structure and characters of irreducible modules are determined, and Loewy diagrams for all Verma and projective modules are found.