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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2508.18218 |
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| _version_ | 1866912554380427264 |
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| author | Mandal, Arunava Singh, Shashank Vikram |
| author_facet | Mandal, Arunava Singh, Shashank Vikram |
| contents | For a class of groups $G$ over a field $\mathbb{F}$, including certain Lie groups, Algebraic groups and finite groups, we develop a general method to determine rational and real elements, thereby unifying earlier group-specific results into a wider framework. As an application, we classify all real and rational elements in the semidirect product ${\rm SL}(2,\mathbb{R}) \ltimes \mathrm{Sym}^n(\mathbb{R}^2)$. Furthermore, for affine groups of the form ${\rm GL}(n,\mathbb{R}) \ltimes \mathbb{R}^n$, we show that if $x \in {\rm GL}(n,\mathbb{R})$ is rational, then $(x,v)$ is rational for every $v \in \mathbb{R}^n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_18218 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On rational and real elements in a class of Lie groups Mandal, Arunava Singh, Shashank Vikram Group Theory Primary 20H20, 20E45, Secondary 20D10, 20G20 For a class of groups $G$ over a field $\mathbb{F}$, including certain Lie groups, Algebraic groups and finite groups, we develop a general method to determine rational and real elements, thereby unifying earlier group-specific results into a wider framework. As an application, we classify all real and rational elements in the semidirect product ${\rm SL}(2,\mathbb{R}) \ltimes \mathrm{Sym}^n(\mathbb{R}^2)$. Furthermore, for affine groups of the form ${\rm GL}(n,\mathbb{R}) \ltimes \mathbb{R}^n$, we show that if $x \in {\rm GL}(n,\mathbb{R})$ is rational, then $(x,v)$ is rational for every $v \in \mathbb{R}^n$. |
| title | On rational and real elements in a class of Lie groups |
| topic | Group Theory Primary 20H20, 20E45, Secondary 20D10, 20G20 |
| url | https://arxiv.org/abs/2508.18218 |