$5 \times 5$-graded Lie algebras, cubic norm structures and quadrangular algebras
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912327529398272 |
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| author | De Medts, Tom Meulewaeter, Jeroen |
| author_facet | De Medts, Tom Meulewaeter, Jeroen |
| contents | We study simple Lie algebras generated by extremal elements, over arbitrary fields of arbitrary characteristic. We show: (1) If the extremal geometry contains lines, then the Lie algebra admits a $5 \times 5$-grading that can be parametrized by a cubic norm structure; (2) If there exists a field extension of degree at most $2$ such that the extremal geometry over that field extension contains lines, and in addition, there exist symplectic pairs of extremal elements, then the Lie algebra admits a $5 \times 5$-grading that can be parametrized by a quadrangular algebra.
One of our key tools is a new definition of exponential maps that makes sense even over fields of characteristic $2$ and $3$, which ought to be interesting in its own right. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_10672 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $5 \times 5$-graded Lie algebras, cubic norm structures and quadrangular algebras De Medts, Tom Meulewaeter, Jeroen Rings and Algebras Group Theory 17B05, 17B20, 17B25, 17B40, 17B45, 17B60, 17C40, 17A30, 20G15, 20G41, 51A50 We study simple Lie algebras generated by extremal elements, over arbitrary fields of arbitrary characteristic. We show: (1) If the extremal geometry contains lines, then the Lie algebra admits a $5 \times 5$-grading that can be parametrized by a cubic norm structure; (2) If there exists a field extension of degree at most $2$ such that the extremal geometry over that field extension contains lines, and in addition, there exist symplectic pairs of extremal elements, then the Lie algebra admits a $5 \times 5$-grading that can be parametrized by a quadrangular algebra. One of our key tools is a new definition of exponential maps that makes sense even over fields of characteristic $2$ and $3$, which ought to be interesting in its own right. |
| title | $5 \times 5$-graded Lie algebras, cubic norm structures and quadrangular algebras |
| topic | Rings and Algebras Group Theory 17B05, 17B20, 17B25, 17B40, 17B45, 17B60, 17C40, 17A30, 20G15, 20G41, 51A50 |
| url | https://arxiv.org/abs/2407.10672 |