Computation of the component group of an arbitrary real algebraic group
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866912055831822336 |
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| author | Timashev, Dmitry A. |
| author_facet | Timashev, Dmitry A. |
| contents | We compute explicitly the group of connected components $π_0G(\mathbb{R})$ of the real Lie group $G(\mathbb{R})$ for an arbitrary (not necessarily linear) connected algebraic group $G$ defined over the field $\mathbb{R}$ of real numbers. In particular, it turns out that $π_0G(\mathbb{R})$ is always an elementary Abelian 2-group. The result looks most transparent in the cases where $G$ is a linear algebraic group or an Abelian variety. The computation is based on structure results on algebraic groups and Galois cohomology methods. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_05214 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Computation of the component group of an arbitrary real algebraic group Timashev, Dmitry A. Group Theory Algebraic Geometry Differential Geometry 14L40, 22E15 (Primary) 11E72, 20G20, 14K20 (Secondary) We compute explicitly the group of connected components $π_0G(\mathbb{R})$ of the real Lie group $G(\mathbb{R})$ for an arbitrary (not necessarily linear) connected algebraic group $G$ defined over the field $\mathbb{R}$ of real numbers. In particular, it turns out that $π_0G(\mathbb{R})$ is always an elementary Abelian 2-group. The result looks most transparent in the cases where $G$ is a linear algebraic group or an Abelian variety. The computation is based on structure results on algebraic groups and Galois cohomology methods. |
| title | Computation of the component group of an arbitrary real algebraic group |
| topic | Group Theory Algebraic Geometry Differential Geometry 14L40, 22E15 (Primary) 11E72, 20G20, 14K20 (Secondary) |
| url | https://arxiv.org/abs/2311.05214 |