Groups with $\mathsf{BC}_\ell$-commutator relations
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866918486403448832 |
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| author | Voronetsky, Egor |
| author_facet | Voronetsky, Egor |
| contents | Isotropic odd unitary groups generalize Chevalley groups of classical types over commutative rings and their twisted forms. Such groups have root subgroups parameterized by a root system $\mathsf{BC}_\ell$ and may be constructed by so-called odd form rings with Peirce decompositions. We show the converse: if a group $G$ has root subgroups indexed by roots of $\mathsf{BC}_\ell$ and satisfying natural conditions, then there is a homomorphism $\mathrm{StU}(R, Δ) \to G$ inducing isomorphisms on the root subgroups, where $\mathrm{StU}(R, Δ)$ is the odd unitary Steinberg group constructed by an odd form ring $(R, Δ)$ with a Peirce decomposition. For groups with root subgroups indexed by $\mathsf A_\ell$ (the already known case) the resulting odd form ring is essentially a generalized matrix ring. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_01225 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Groups with $\mathsf{BC}_\ell$-commutator relations Voronetsky, Egor Group Theory 20G99 Isotropic odd unitary groups generalize Chevalley groups of classical types over commutative rings and their twisted forms. Such groups have root subgroups parameterized by a root system $\mathsf{BC}_\ell$ and may be constructed by so-called odd form rings with Peirce decompositions. We show the converse: if a group $G$ has root subgroups indexed by roots of $\mathsf{BC}_\ell$ and satisfying natural conditions, then there is a homomorphism $\mathrm{StU}(R, Δ) \to G$ inducing isomorphisms on the root subgroups, where $\mathrm{StU}(R, Δ)$ is the odd unitary Steinberg group constructed by an odd form ring $(R, Δ)$ with a Peirce decomposition. For groups with root subgroups indexed by $\mathsf A_\ell$ (the already known case) the resulting odd form ring is essentially a generalized matrix ring. |
| title | Groups with $\mathsf{BC}_\ell$-commutator relations |
| topic | Group Theory 20G99 |
| url | https://arxiv.org/abs/2308.01225 |