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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Académie des sciences
2025-03-01
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| Series: | Comptes Rendus. Mathématique |
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| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.702/ |
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| author | Jouteur, Perrine |
| author_facet | Jouteur, Perrine |
| author_sort | Jouteur, Perrine |
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| description | 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. |
| format | Article |
| id | doaj-art-c6112e1923ba4292aadf45ba1944444e |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2025-03-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-c6112e1923ba4292aadf45ba1944444e2025-08-20T03:58:11ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692025-03-01363G18910710.5802/crmath.70210.5802/crmath.702Burau representation of $B_4$ and quantization of the rational projective planeJouteur, Perrine0Laboratoire de Mathématiques de Reims, UMR 9008 CNRS et Université de Reims Champagne-Ardenne, U.F.R. Sciences Exactes et Naturelles, Moulin de la Housse, BP 1039, 51687 Reims cedex 2, FranceThe 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.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.702/QuantizationBurau representation$q$-rational numbersbraid grouprational projective plane |
| spellingShingle | Jouteur, Perrine Burau representation of $B_4$ and quantization of the rational projective plane Comptes Rendus. Mathématique Quantization Burau representation $q$-rational numbers braid group rational projective plane |
| title | Burau representation of $B_4$ and quantization of the rational projective plane |
| title_full | Burau representation of $B_4$ and quantization of the rational projective plane |
| title_fullStr | Burau representation of $B_4$ and quantization of the rational projective plane |
| title_full_unstemmed | Burau representation of $B_4$ and quantization of the rational projective plane |
| title_short | Burau representation of $B_4$ and quantization of the rational projective plane |
| title_sort | burau representation of b 4 and quantization of the rational projective plane |
| topic | Quantization Burau representation $q$-rational numbers braid group rational projective plane |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.702/ |
| work_keys_str_mv | AT jouteurperrine buraurepresentationofb4andquantizationoftherationalprojectiveplane |