$\mathbf {5 \times 5}$ -graded Lie algebras, cubic norm structures and quadrangular algebras
We study simple Lie algebras generated by extremal elements, over arbitrary fields of arbitrary characteristic. We show the following: (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 the...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Cambridge University Press
2025-01-01
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| Series: | Forum of Mathematics, Sigma |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509425000465/type/journal_article |
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| Summary: | We study simple Lie algebras generated by extremal elements, over arbitrary fields of arbitrary characteristic. We show the following: (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. |
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| ISSN: | 2050-5094 |