The Drinfeld Center of the Generic Temperley--Lieb Category
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911555994517504 |
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| author | Alqady, Moaaz |
| author_facet | Alqady, Moaaz |
| contents | 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 $q$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_28970 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Drinfeld Center of the Generic Temperley--Lieb Category Alqady, Moaaz Quantum Algebra Representation Theory 18M15 (Primary) 17B37, 03C20 (Secondary) 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 $q$. |
| title | The Drinfeld Center of the Generic Temperley--Lieb Category |
| topic | Quantum Algebra Representation Theory 18M15 (Primary) 17B37, 03C20 (Secondary) |
| url | https://arxiv.org/abs/2603.28970 |