A Coboundary Temperely-Lieb Category for $\mathfrak{sl}_2$-Crystals
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915176288092160 |
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| author | Alqady, Moaaz Stroiński, Mateusz |
| author_facet | Alqady, Moaaz Stroiński, Mateusz |
| contents | By considering a suitable renormalization of the Temperley--Lieb category, we study its specialization to the case $q=0$. Unlike the $q\neq 0$ case, the obtained monoidal category, $\mathcal{TL}_0(\Bbbk)$, is not rigid or braided. We provide a closed formula for the Jones--Wenzl projectors in $\mathcal{TL}_0(\Bbbk)$ and give semisimple bases for its endomorphism algebras. We explain how to obtain the same basis using the representation theory of finite inverse monoids, via the associated Möbius inversion. We then describe a coboundary structure on $\mathcal{TL}_0(\Bbbk)$ and show that its idempotent completion is coboundary monoidally equivalent to the category of $\mathfrak{sl}_{2}$-crystals. This gives a diagrammatic description of the commutor for $\mathfrak{sl}_{2}$-crystals defined by Henriques and Kamnitzer and of the resulting action of the cactus group. We also study fiber functors of $\mathcal{TL}_0(\Bbbk)$ and discuss how they differ from the $q\neq 0$ case. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_05732 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Coboundary Temperely-Lieb Category for $\mathfrak{sl}_2$-Crystals Alqady, Moaaz Stroiński, Mateusz Representation Theory Combinatorics Quantum Algebra 17B37 (Primary) 05E10 (Secondary) By considering a suitable renormalization of the Temperley--Lieb category, we study its specialization to the case $q=0$. Unlike the $q\neq 0$ case, the obtained monoidal category, $\mathcal{TL}_0(\Bbbk)$, is not rigid or braided. We provide a closed formula for the Jones--Wenzl projectors in $\mathcal{TL}_0(\Bbbk)$ and give semisimple bases for its endomorphism algebras. We explain how to obtain the same basis using the representation theory of finite inverse monoids, via the associated Möbius inversion. We then describe a coboundary structure on $\mathcal{TL}_0(\Bbbk)$ and show that its idempotent completion is coboundary monoidally equivalent to the category of $\mathfrak{sl}_{2}$-crystals. This gives a diagrammatic description of the commutor for $\mathfrak{sl}_{2}$-crystals defined by Henriques and Kamnitzer and of the resulting action of the cactus group. We also study fiber functors of $\mathcal{TL}_0(\Bbbk)$ and discuss how they differ from the $q\neq 0$ case. |
| title | A Coboundary Temperely-Lieb Category for $\mathfrak{sl}_2$-Crystals |
| topic | Representation Theory Combinatorics Quantum Algebra 17B37 (Primary) 05E10 (Secondary) |
| url | https://arxiv.org/abs/2502.05732 |