Edifices: Building-like spaces associated to linear algebraic groups
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866913325487489024 |
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| author | Bate, Michael Martin, Benjamin Roehrle, Gerhard |
| author_facet | Bate, Michael Martin, Benjamin Roehrle, Gerhard |
| contents | Given a semisimple linear algebraic $k$-group $G$, one has a spherical building $Δ_G$, and one can interpret the geometric realisation $Δ_G(\mathbb R)$ of $Δ_G$ in terms of cocharacters of $G$. The aim of this paper is to extend this construction to the case when $G$ is an arbitrary connected linear algebraic group; we call the resulting object $Δ_G(\mathbb R)$ the spherical edifice of $G$. We also define an object $V_G(\mathbb R)$ which is an analogue of the vector building for a semisimple group; we call $V_G(\mathbb R)$ the vector edifice. The notions of a linear map and an isomorphism between edifices are introduced; we construct some linear maps arising from natural group-theoretic operations. We also devise a family of metrics on $V_G(\mathbb R)$ and show they are all bi-Lipschitz equivalent to each other; with this extra structure, $V_G(\mathbb R)$ becomes a complete metric space. Finally, we present some motivation in terms of geometric invariant theory and variations on the Tits Centre Conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_11770 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Edifices: Building-like spaces associated to linear algebraic groups Bate, Michael Martin, Benjamin Roehrle, Gerhard Group Theory Metric Geometry 51E24, 20E42, 20G15 Given a semisimple linear algebraic $k$-group $G$, one has a spherical building $Δ_G$, and one can interpret the geometric realisation $Δ_G(\mathbb R)$ of $Δ_G$ in terms of cocharacters of $G$. The aim of this paper is to extend this construction to the case when $G$ is an arbitrary connected linear algebraic group; we call the resulting object $Δ_G(\mathbb R)$ the spherical edifice of $G$. We also define an object $V_G(\mathbb R)$ which is an analogue of the vector building for a semisimple group; we call $V_G(\mathbb R)$ the vector edifice. The notions of a linear map and an isomorphism between edifices are introduced; we construct some linear maps arising from natural group-theoretic operations. We also devise a family of metrics on $V_G(\mathbb R)$ and show they are all bi-Lipschitz equivalent to each other; with this extra structure, $V_G(\mathbb R)$ becomes a complete metric space. Finally, we present some motivation in terms of geometric invariant theory and variations on the Tits Centre Conjecture. |
| title | Edifices: Building-like spaces associated to linear algebraic groups |
| topic | Group Theory Metric Geometry 51E24, 20E42, 20G15 |
| url | https://arxiv.org/abs/2305.11770 |