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The Drinfeld Center of the Generic Temperley--Lieb Category

Moaaz Alqady

Abstract

We show that the Temperley--Lieb category $\mathbf{TL}(q;\mathbb{C})$ embeds in an ultraproduct of modular tensor categories when $q$ is not a root of unity. As a result, we show that its Drinfeld center is semisimple and describe its simple objects. The canonical functor $$\mathbf{TL}(q;\mathbb{C})\boxtimes \mathbf{TL}(q;\mathbb{C})^{\mathrm{rev}} \boxtimes \mathbf{Rep}(\mathbb{Z}/2\mathbb{Z}) \to \mathcal Z(\mathbf{TL}(q;\mathbb{C})),$$ induced by the braiding and the $\mathbb{Z}/2\mathbb{Z}$--grading on the Temperley--Lieb category, is thus shown to be a monoidal equivalence, which becomes a braided equivalence upon twisting the braiding by a certain bicharacter. Along the way, we formalize some general results about ultraproducts of tensor categories and tensor functors, building on earlier works of Crumley, Harman, and Flake--Harman--Laugwitz. We also discuss the center at some exceptional values of $q$.

The Drinfeld Center of the Generic Temperley--Lieb Category

Abstract

We show that the Temperley--Lieb category embeds in an ultraproduct of modular tensor categories when is not a root of unity. As a result, we show that its Drinfeld center is semisimple and describe its simple objects. The canonical functor induced by the braiding and the --grading on the Temperley--Lieb category, is thus shown to be a monoidal equivalence, which becomes a braided equivalence upon twisting the braiding by a certain bicharacter. Along the way, we formalize some general results about ultraproducts of tensor categories and tensor functors, building on earlier works of Crumley, Harman, and Flake--Harman--Laugwitz. We also discuss the center at some exceptional values of .

Paper Structure

This paper contains 7 sections, 65 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on Tensor Categories
  3. The Temperley--Lieb Category
  4. ${\mathrm{\bf{TL}}}\left(q;\, \mathbb C\right)$ in an Ultraproduct of Fusion Categories
  5. The Center of the Generic Temperley--Lieb Category
  6. Some Exceptional Cases and Concluding Remarks
  7. Ultraproducts for the Working $\otimes$-Categorician

Key Result

Proposition 2.5

GN Let $\mathcal{C}$ be a tensor category with a faithful $G$ grading. Then, we have an injective group homomorphism from $G^\vee:=\mathrm{Hom}(G,\mathbb K^\times)$ to the group of half-braidings (under $\otimes$) on $\mathds 1_\mathcal{C}$ in $\mathcal{C}$. When $G$ is abelian, this induces a funct

Theorems & Definitions (152)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • Remark 2.7
  • Definition 2.8
  • Definition 2.9
  • ...and 142 more