On stable Cartan subgroups of Lie groups
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908442390691840 |
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| author | Kumar, Parteek Mandal, Arunava Singh, Shashank Vikram |
| author_facet | Kumar, Parteek Mandal, Arunava Singh, Shashank Vikram |
| contents | Let $G$ be a connected real Lie group with associated Lie algebra $\mathfrak g$, and let ${\rm Aut}(G)$ be the group of (Lie) automorphisms of $G$. It is noted here that, given a super-solvable subgroup $Γ\subset {\rm Aut}(G)$ of semisimple automorphisms, there exists a $Γ$-stable Cartan subgroup, by using a result of Borel and Mostow. We characterize the $Γ$-stable Cartan subgroups (with induced action) in the quotient group modulo a $Γ$-stable closed normal subgroup as the images of the $Γ$-stable Cartan subgroups in the ambient group. It is well known that a semisimple automorphism of $\mathfrak g$ always fixes a Cartan subalgebra of $\mathfrak g$. Conversely, if we take a representative from each non-conjugate class of Cartan subalgebras in a real Lie algebra, we show that there exists a non-identity automorphism that fixes these representatives. We explicitly identify such automorphisms in the case of classical simple Lie algebras. As a consequence, we deduce an analogous result for semisimple Lie groups. Moreover, given a $Γ$-stable Cartan subgroup $H$ of $G$, and a $Γ$-stable closed connected normal subgroup $M$ of $G$, we prove that there exists a $Γ$-stable Cartan subgroup $H_M$ of $M$ such that $H\cap M\subset H_M$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_07027 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On stable Cartan subgroups of Lie groups Kumar, Parteek Mandal, Arunava Singh, Shashank Vikram Group Theory 22E15, 22E25 Let $G$ be a connected real Lie group with associated Lie algebra $\mathfrak g$, and let ${\rm Aut}(G)$ be the group of (Lie) automorphisms of $G$. It is noted here that, given a super-solvable subgroup $Γ\subset {\rm Aut}(G)$ of semisimple automorphisms, there exists a $Γ$-stable Cartan subgroup, by using a result of Borel and Mostow. We characterize the $Γ$-stable Cartan subgroups (with induced action) in the quotient group modulo a $Γ$-stable closed normal subgroup as the images of the $Γ$-stable Cartan subgroups in the ambient group. It is well known that a semisimple automorphism of $\mathfrak g$ always fixes a Cartan subalgebra of $\mathfrak g$. Conversely, if we take a representative from each non-conjugate class of Cartan subalgebras in a real Lie algebra, we show that there exists a non-identity automorphism that fixes these representatives. We explicitly identify such automorphisms in the case of classical simple Lie algebras. As a consequence, we deduce an analogous result for semisimple Lie groups. Moreover, given a $Γ$-stable Cartan subgroup $H$ of $G$, and a $Γ$-stable closed connected normal subgroup $M$ of $G$, we prove that there exists a $Γ$-stable Cartan subgroup $H_M$ of $M$ such that $H\cap M\subset H_M$. |
| title | On stable Cartan subgroups of Lie groups |
| topic | Group Theory 22E15, 22E25 |
| url | https://arxiv.org/abs/2507.07027 |