Burau representation of $B_4$ and quantization of the rational projective plane
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913935420030976 |
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| author | Jouteur, Perrine |
| author_facet | Jouteur, Perrine |
| contents | 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 $\mathbb{P}^2(\mathbb{Z}(q))$, 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 $\mathbb{P}^2(\mathbb{Z}(q))$, and show existence of embeddings of the $q$-deformed projective line $\mathbb{P}^1(\mathbb{Z}(q))$ that precisely correspond to the notion of $q$-rationals due to Morier-Genoud and Ovsienko. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_20645 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Burau representation of $B_4$ and quantization of the rational projective plane Jouteur, Perrine Group Theory 20F36, 20C12, 05A30 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 $\mathbb{P}^2(\mathbb{Z}(q))$, 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 $\mathbb{P}^2(\mathbb{Z}(q))$, and show existence of embeddings of the $q$-deformed projective line $\mathbb{P}^1(\mathbb{Z}(q))$ that precisely correspond to the notion of $q$-rationals due to Morier-Genoud and Ovsienko. |
| title | Burau representation of $B_4$ and quantization of the rational projective plane |
| topic | Group Theory 20F36, 20C12, 05A30 |
| url | https://arxiv.org/abs/2407.20645 |