Ribbon categories from ind-exact algebras: simple current case

TL;DR

This paper establishes criteria for local modules over certain algebras in braided tensor categories to inherit ribbon structures, and applies this to simple current algebras to produce new ribbon categories.

Contribution

It provides new conditions under which local modules over simple current algebras form ribbon tensor categories, with applications to quantum supergroup categories.

Findings

Criteria for local modules to inherit ribbon structures.

Verification of hypotheses for simple current algebras.

Construction of ribbon categories from quantum supergroups.

Abstract

We give criteria for when finitely generated local modules over a commutative algebra $A$ in the ind-completion $\widehat{\mathcal{C}}$ of a braided tensor category $\mathcal{C}$ inherit the structure of a (rigid, braided, ribbon) tensor category. We then apply this to simple current algebras $A = \bigoplus_{g \in \Gamma} E_g$, where $\Gamma$ is a subgroup of invertible objects in $\mathcal{C}$. Using a description of simple $A$-modules, we verify the required hypotheses for this class of algebras and deduce rigidity, braided, ribbon, and non-degeneracy properties for their finitely generated local modules. As applications, we construct examples of ribbon tensor categories from quantum supergroup categories for unrolled $\mathfrak{gl}(1|1)$.