Symmetries of the q-deformed real projective line
Fuente:
arXiv
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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909524066041856 |
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| author | Jouteur, Perrine |
| author_facet | Jouteur, Perrine |
| contents | We generalize in two steps the quantized action of the modular group on $q$-deformed real numbers introduced by Morier-Genoud and Ovsienko. First, we let the projective general linear group $PGL_2(\mathbb{Z})$ act on $q$-real numbers via a $q$-deformed action. The quantized matrices we get have combinatorial interpretations. Then we consider an extension of the group $PGL_2(\mathbb{Z})$ by the $2$-elements cyclic group, and define a quantized action of this extension on $q$-real numbers. We deduce from these actions some underlying relations between $q$-real numbers, and between left and right versions of $q$-deformed rational numbers. In particular we investigate the case of some algebraic numbers of degree $4$ and $6$. We also prove that the way of quantizing real numbers defined by Morier-Genoud and Ovsienko is an injective process. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_02122 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Symmetries of the q-deformed real projective line Jouteur, Perrine Combinatorics We generalize in two steps the quantized action of the modular group on $q$-deformed real numbers introduced by Morier-Genoud and Ovsienko. First, we let the projective general linear group $PGL_2(\mathbb{Z})$ act on $q$-real numbers via a $q$-deformed action. The quantized matrices we get have combinatorial interpretations. Then we consider an extension of the group $PGL_2(\mathbb{Z})$ by the $2$-elements cyclic group, and define a quantized action of this extension on $q$-real numbers. We deduce from these actions some underlying relations between $q$-real numbers, and between left and right versions of $q$-deformed rational numbers. In particular we investigate the case of some algebraic numbers of degree $4$ and $6$. We also prove that the way of quantizing real numbers defined by Morier-Genoud and Ovsienko is an injective process. |
| title | Symmetries of the q-deformed real projective line |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2503.02122 |