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The braided monoidal structure of tube algebra representations

David Jaklitsch, Makoto Yamashita

Abstract

We consider the tube algebra of a spherical semisimple multitensor category $\mathcal{X}$, and construct a braided monoidal structure with twist for its representations. We further show that this category is braided tensor equivalent with the Drinfeld center of the ind-category of $\mathcal{X}$, extending the well-known linear equivalence.

The braided monoidal structure of tube algebra representations

Abstract

We consider the tube algebra of a spherical semisimple multitensor category , and construct a braided monoidal structure with twist for its representations. We further show that this category is braided tensor equivalent with the Drinfeld center of the ind-category of , extending the well-known linear equivalence.

Paper Structure

This paper contains 18 sections, 24 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Preliminaries
  4. Spherical multitensor categories
  5. Ind-completion
  6. Drinfeld center
  7. Algebras with local units
  8. Graphical calculus and spaces of tubes
  9. Cyclically invariant vector spaces of morphisms
  10. Diagrammatic calculus
  11. Spherical pairing
  12. Spaces of tubes and the Tube algebra
  13. Tube algebra representations
  14. Tensor product of representations
  15. Associators and pentagon axiom
  16. ...and 3 more sections

Key Result

Lemma 3.4

Let $\mathcal{X}$ be a pivotal multitensor category.

Theorems & Definitions (69)

  • Definition 2.1: cf. MR1966524MR3342166
  • Remark 2.2
  • Definition 2.3: MR695890MR899719
  • Definition 2.4
  • Definition 3.1
  • Remark 3.2
  • Remark 3.3
  • Lemma 3.4
  • proof
  • Definition 3.5
  • ...and 59 more