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Ribbon categories from ind-exact algebras: simple current case

Kenichi Shimizu, Harshit Yadav

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 $A = \bigoplus_{g \in Γ} E_g$, where $Γ$ 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 $\mathfrak{gl}(1|1)$.

Ribbon categories from ind-exact algebras: simple current case

Abstract

We give criteria for when finitely generated local modules over a commutative algebra in the ind-completion of a braided tensor category inherit the structure of a (rigid, braided, ribbon) tensor category. We then apply this to simple current algebras , where is a subgroup of invertible objects in . Using a description of simple -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 .

Paper Structure

This paper contains 72 sections, 64 theorems, 149 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. VOA motivation and applications
  4. Examples
  5. Organization
  6. Acknowledgements
  7. Background
  8. Notations
  9. Tensor categories
  10. Braided tensor categories
  11. Müger center and nondegeneracy
  12. Ribbon tensor categories
  13. Modules in a closed monoidal category
  14. Modules over a central algebra
  15. Local modules
  16. ...and 57 more sections

Key Result

Theorem 1

Let $A\in\widehat{\mathcal{C}}$ be a haploid commutative algebra.

Figures (4)

  • Figure 1: Proof of Equation \ref{['eq:eta-X-coboundary-condition']}
  • Figure 2: Proof of Lemma \ref{['lem:module-functor-FX']}
  • Figure 3: Proof of Theorem \ref{['thm:classification-2']}
  • Figure 4: Proof of the zig-zag equation

Theorems & Definitions (131)

  • Theorem : A
  • Theorem : B
  • Theorem : C
  • Lemma 2.1: deligne2002categories
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 121 more