Collineation groups of octonionic and split-octonionic planes
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866909196455247872 |
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| author | Corradetti, Daniele Marrani, Alessio Zucconi, Francesco |
| author_facet | Corradetti, Daniele Marrani, Alessio Zucconi, Francesco |
| contents | We present a Veronese formulation of the octonionic and split-octonionic projective and hyperbolic planes. This formulation of the incidence planes highlights the relationship between the Veronese vectors and the rank-1 elements of the Albert algebras over octonions and split-octonions, yielding to a clear formulation of the relationship with the real forms of the Lie groups arising as collineation groups of these planes. The Veronesean representation also provides a novel and minimal construction of the same octonionic and split-octonionic planes, by exploiting two symmetric composition algebras: the Okubo algebra and the paraoctonionic algebra. Besides the intrinsic mathematical relevance of this construction of the real forms of the Cayley-Moufang plane, we expect this approach to have implications in all mathematical physics related with exceptional Lie Groups of type $G_{2},F_{4}$ and $E_{6}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_11907 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Collineation groups of octonionic and split-octonionic planes Corradetti, Daniele Marrani, Alessio Zucconi, Francesco Rings and Algebras Mathematical Physics 17A35, 51M35, 17B25 We present a Veronese formulation of the octonionic and split-octonionic projective and hyperbolic planes. This formulation of the incidence planes highlights the relationship between the Veronese vectors and the rank-1 elements of the Albert algebras over octonions and split-octonions, yielding to a clear formulation of the relationship with the real forms of the Lie groups arising as collineation groups of these planes. The Veronesean representation also provides a novel and minimal construction of the same octonionic and split-octonionic planes, by exploiting two symmetric composition algebras: the Okubo algebra and the paraoctonionic algebra. Besides the intrinsic mathematical relevance of this construction of the real forms of the Cayley-Moufang plane, we expect this approach to have implications in all mathematical physics related with exceptional Lie Groups of type $G_{2},F_{4}$ and $E_{6}$. |
| title | Collineation groups of octonionic and split-octonionic planes |
| topic | Rings and Algebras Mathematical Physics 17A35, 51M35, 17B25 |
| url | https://arxiv.org/abs/2311.11907 |