The Drinfeld Center of the Generic Temperley--Lieb Category

TL;DR

This paper demonstrates that the Temperley--Lieb category embeds into an ultraproduct of modular tensor categories for generic q, and characterizes its Drinfeld center as semisimple with explicit simple objects.

Contribution

It establishes a monoidal and braided equivalence for the Drinfeld center of the Temperley--Lieb category at generic q, extending the understanding of its categorical structure.

Findings

The Drinfeld center of the Temperley--Lieb category is semisimple.

The embedding into an ultraproduct of modular tensor categories is shown for generic q.

The monoidal and braided equivalences are explicitly described.

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 $q$.