Kac-Moody Symmetric Spaces: arbitrary symmetrizable complex or almost split real type
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914054667239424 |
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| author | Köhl, Ralf Vock, Christian |
| author_facet | Köhl, Ralf Vock, Christian |
| contents | Kac-Moody symmetric spaces have been introduced by Freyn, Hartnick, Horn and the first-named author for centered Kac-Moody groups, that is, Kac-Moody groups that are generated by their root subgroups. In the case of non-invertible generalized Cartan matrices this leads to complications that -- within the approach proposed originally -- cannot be repaired in the affine case. In the present article we propose an alternative approach to Kac-Moody symmetric spaces which for invertible generalized Cartan matrices provides exactly the same concept, which for the non-affine non-invertible case provides alternative Kac-Moody symmetric spaces, and which finally provides Kac-Moody symmetric spaces for affine Kac-Moody groups. In a nutshell, the original intention by Freyn, Hartnick, Horn and Köhl was to construct symmetric spaces that likely lead to primitive actions of the Kac-Moody groups; this, of course, cannot work in the affine case as affine Kac-Moody groups are far from simple. Additionally, we study the Galois descent to almost split real Kac-Moody symmetric spaces based on the theory of almost split Kac-Moody groups developed by Rémy 2002. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_02176 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Kac-Moody Symmetric Spaces: arbitrary symmetrizable complex or almost split real type Köhl, Ralf Vock, Christian Group Theory Kac-Moody symmetric spaces have been introduced by Freyn, Hartnick, Horn and the first-named author for centered Kac-Moody groups, that is, Kac-Moody groups that are generated by their root subgroups. In the case of non-invertible generalized Cartan matrices this leads to complications that -- within the approach proposed originally -- cannot be repaired in the affine case. In the present article we propose an alternative approach to Kac-Moody symmetric spaces which for invertible generalized Cartan matrices provides exactly the same concept, which for the non-affine non-invertible case provides alternative Kac-Moody symmetric spaces, and which finally provides Kac-Moody symmetric spaces for affine Kac-Moody groups. In a nutshell, the original intention by Freyn, Hartnick, Horn and Köhl was to construct symmetric spaces that likely lead to primitive actions of the Kac-Moody groups; this, of course, cannot work in the affine case as affine Kac-Moody groups are far from simple. Additionally, we study the Galois descent to almost split real Kac-Moody symmetric spaces based on the theory of almost split Kac-Moody groups developed by Rémy 2002. |
| title | Kac-Moody Symmetric Spaces: arbitrary symmetrizable complex or almost split real type |
| topic | Group Theory |
| url | https://arxiv.org/abs/2309.02176 |