The Drinfeld Center of the Generic Temperley--Lieb Category

We show that the Temperley–Lieb category $\mathbf{T}\mathbf{L}(q;\mathbb{C})$ 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 $$\mathbf{T}\mathbf{L}(q;\mathbb{C})⊠\mathbf{T}\mathbf{L}(q;\mathbb{C}{)}^{\mathrm{r}\mathrm{e}\mathrm{v}}⊠\mathbf{R}\mathbf{e}\mathbf{p}(\mathbb{Z}\mathrm{/}2\mathbb{Z})\to \mathcal{Z}(\mathbf{T}\mathbf{L}(q;\mathbb{C})),$$ induced by the braiding and the $\mathbb{Z}\mathrm{/}2\mathbb{Z}$–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 $q$.