Centralisers of semi-simple elements are semidirect products
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917414964297728 |
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| author | Digne, François Michel, Jean |
| author_facet | Digne, François Michel, Jean |
| contents | Let $\mathbf G$ be a connected reductive algebraic group over an algebraically closed field, and let $s\in\mathbf G$ be a semisimple element. We show that the centraliser of $s$ is the semi-direct product of its identity component by its group of components. We then look at the case where $\mathbf G$ is defined over an algebraic closure of a finite field ${\mathbb F}_q$, and $F$ is a Frobenius endomorphism attached to an ${\mathbb F}_q$-structure on $\mathbf G$. We show that if the centraliser of $s$ is $F$-stable we have a semi-direct product decomposition of the $F$-fixed points. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_19164 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Centralisers of semi-simple elements are semidirect products Digne, François Michel, Jean Group Theory Let $\mathbf G$ be a connected reductive algebraic group over an algebraically closed field, and let $s\in\mathbf G$ be a semisimple element. We show that the centraliser of $s$ is the semi-direct product of its identity component by its group of components. We then look at the case where $\mathbf G$ is defined over an algebraic closure of a finite field ${\mathbb F}_q$, and $F$ is a Frobenius endomorphism attached to an ${\mathbb F}_q$-structure on $\mathbf G$. We show that if the centraliser of $s$ is $F$-stable we have a semi-direct product decomposition of the $F$-fixed points. |
| title | Centralisers of semi-simple elements are semidirect products |
| topic | Group Theory |
| url | https://arxiv.org/abs/2512.19164 |