Burau representation of $B_4$ and quantization of the rational projective plane
The braid group $B_4$ naturally acts on the rational projective plane $\mathbb{P}^2(\mathbb{Q})$, this action corresponds to the classical integral reduced Burau representation of $B_4$. The first result of this paper is a classification of the orbits of this action. The Burau representation then de...
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| Format: | Article |
| Language: | English |
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Académie des sciences
2025-03-01
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| Series: | Comptes Rendus. Mathématique |
| Subjects: | |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.702/ |
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| Summary: | The braid group $B_4$ naturally acts on the rational projective plane $\mathbb{P}^2(\mathbb{Q})$, this action corresponds to the classical integral reduced Burau representation of $B_4$. The first result of this paper is a classification of the orbits of this action. The Burau representation then defines an action of $B_4$ on $\mathbb{P}^2\bigl (\mathbb{Z}(q)\bigr )$, where $q$ is a formal parameter and $\mathbb{Z}(q)$ is the field of rational functions in $q$ with integer coefficients. We study orbits of the $B_4$-action on $\mathbb{P}^2\bigl (\mathbb{Z}(q)\bigr )$, and show existence of embeddings of the $q$-deformed projective line $\mathbb{P}^1\bigl (\mathbb{Z}(q)\bigr )$ that precisely correspond to the notion of $q$-rationals due to Morier-Genoud and Ovsienko. |
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| ISSN: | 1778-3569 |