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On braided simple extensions and braided non-semisimple near-group categories

Daniel Sebbag

Abstract

We study simple extensions of pointed finite tensor categories, that is, tensor categories $\mathcal{C}$ admitting an abelian decomposition $\mathcal{C} \cong \mathcal{D} \oplus \mathcal{M}$ where $\mathcal{D}$ is a pointed tensor subcategory and $\mathcal{M}$ has a unique simple projective object. Such categories provide a natural generalization of near-group categories. Our results concern the braided case. We prove that every non-degenerate braided non-semisimple near-group category is a braided simple extension of $\mathrm{sRep}(W\oplus W^*)$ with non-trivial braiding for which $\mathrm{sRep}(W)$ is Lagrangian. Moreover, any braided non-semisimple near-group category $\mathcal{C}$ arises canonically as an extension of such a category by $\mathrm{Rep}(G)$, where $G$ is the Picard group of a symmetric subcategory determined by the unique simple projective object of $\mathcal{C}$.

On braided simple extensions and braided non-semisimple near-group categories

Abstract

We study simple extensions of pointed finite tensor categories, that is, tensor categories admitting an abelian decomposition where is a pointed tensor subcategory and has a unique simple projective object. Such categories provide a natural generalization of near-group categories. Our results concern the braided case. We prove that every non-degenerate braided non-semisimple near-group category is a braided simple extension of with non-trivial braiding for which is Lagrangian. Moreover, any braided non-semisimple near-group category arises canonically as an extension of such a category by , where is the Picard group of a symmetric subcategory determined by the unique simple projective object of .

Paper Structure

This paper contains 15 sections, 20 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Structure of the paper
  3. Preliminaries
  4. Notation and terminology
  5. Braiding, symmetry, twists and Müger centralizer
  6. De-equivariantization
  7. Braided exact sequences of tensor categories
  8. The structure of braided simple extensions
  9. Basic properties of simple extensions
  10. Degeneracy in braided simple extensions
  11. Braided simple extensions of pointed tensor categories
  12. The Q-braiding group of C
  13. Non-semisimple braided near-group categories
  14. De-equivariantization of near-group categories
  15. Non-degenerate near-group categories

Key Result

Theorem 1

Let $\mathcal{C}$ be a non-semisimple braided near-group category with generalized fusion rule $(G,r)$, then $r=0$, and consequently $\mathcal{C}$ is weakly integral.

Theorems & Definitions (48)

  • Definition 1.0.2
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 4
  • Theorem 5
  • Proposition 2.3.1
  • proof
  • Remark 2.3.2
  • Proposition 2.3.3
  • ...and 38 more