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$G$-crossed braided zesting

Colleen Delaney, César Galindo, Julia Plavnik, Eric Rowell, Qing Zhang

Abstract

For a finite group $G$, a $G$-crossed braided fusion category is $G$-graded fusion category with additional structures, namely a $G$-action and a $G$-braiding. We develop the notion of $G$-crossed braided zesting: an explicit method for constructing new $G$-crossed braided fusion categories from a given one by means of cohomological data associated with the invertible objects in the category and grading group $G$. This is achieved by adapting a similar construction for (braided) fusion categories recently described by the authors. All $G$-crossed braided zestings of a given category $\mathcal{C}$ are $G$-extensions of their trivial component and can be interpreted in terms of the homotopy-based description of Etingof, Nikshych and Ostrik. In particular, we explicitly describe which $G$-extensions correspond to $G$-crossed braided zestings.

$G$-crossed braided zesting

Abstract

For a finite group , a -crossed braided fusion category is -graded fusion category with additional structures, namely a -action and a -braiding. We develop the notion of -crossed braided zesting: an explicit method for constructing new -crossed braided fusion categories from a given one by means of cohomological data associated with the invertible objects in the category and grading group . This is achieved by adapting a similar construction for (braided) fusion categories recently described by the authors. All -crossed braided zestings of a given category are -extensions of their trivial component and can be interpreted in terms of the homotopy-based description of Etingof, Nikshych and Ostrik. In particular, we explicitly describe which -extensions correspond to -crossed braided zestings.
Paper Structure (21 sections, 9 theorems, 71 equations, 1 table)

This paper contains 21 sections, 9 theorems, 71 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. G-crossed braided tensor categories
  4. Associative and braided zesting
  5. Diagrammatic summary of the braided zesting construction
  6. G-crossed zesting
  7. Associative G-crossed zesting
  8. Zested G-crossed braiding and G-action
  9. From Central G-extensions to G-crossed braided tensor categories
  10. Zesting central functors
  11. Connection between G-crossed zesting and G-extension theory
  12. Z/NZ-crossed zesting
  13. Recovering the braided zesting construction
  14. Zested rigidity, pivotality, and trace
  15. Rigidity of G-crossed associative zesting
  16. ...and 6 more sections

Key Result

Lemma 2.9

The $G$-equivariantizations ${\mathcal{C}}^G,\mathcal{D}^G$ of two equivalent $G$-crossed braided categories ${\mathcal{C}},\mathcal{D}$ are equivalent as braided fusion categories.

Theorems & Definitions (51)

  • Definition 2.1
  • Example 2.2
  • Remark 2.3
  • Example 2.4
  • Remark 2.5
  • Definition 2.6: Adapted from Section 3.1 of Galindo2017
  • Definition 2.7
  • Remark 2.8
  • Lemma 2.9
  • proof
  • ...and 41 more