Groups with $\mathsf A_\ell$-commutator relations
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arXiv
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866909018294845440 |
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| author | Voronetsky, Egor |
| author_facet | Voronetsky, Egor |
| contents | If $A$ is a unital associative ring and $\ell \geq 2$, then the general linear group $\mathrm{GL}(\ell, A)$ has root subgroups $U_α$ and Weyl elements $n_α$ for $α$ from the root system of type $\mathsf A_{\ell - 1}$. Conversely, if an arbitrary group has such root subgroups and Weyl elements for $\ell \geq 4$ satisfying natural conditions, then there is a way to recover the ring $A$. We prove a generalization of this result not using the Weyl elements, so instead of the matrix ring $\mathrm M(\ell, A)$ we construct a non-unital associative ring with a well-behaved Peirce decomposition. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_16182 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Groups with $\mathsf A_\ell$-commutator relations Voronetsky, Egor Group Theory 20G99 If $A$ is a unital associative ring and $\ell \geq 2$, then the general linear group $\mathrm{GL}(\ell, A)$ has root subgroups $U_α$ and Weyl elements $n_α$ for $α$ from the root system of type $\mathsf A_{\ell - 1}$. Conversely, if an arbitrary group has such root subgroups and Weyl elements for $\ell \geq 4$ satisfying natural conditions, then there is a way to recover the ring $A$. We prove a generalization of this result not using the Weyl elements, so instead of the matrix ring $\mathrm M(\ell, A)$ we construct a non-unital associative ring with a well-behaved Peirce decomposition. |
| title | Groups with $\mathsf A_\ell$-commutator relations |
| topic | Group Theory 20G99 |
| url | https://arxiv.org/abs/2203.16182 |