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arXiv · 2407.10672

$5 \times 5$-graded Lie algebras, cubic norm structures and quadrangular algebras

Tom De Medts·Jeroen Meulewaeter
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Abstract

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.

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Tom De Medts, Jeroen Meulewaeter. 2024-07-15. $5 \times 5$-graded Lie algebras, cubic norm structures and quadrangular algebras. https://arxiv.org/abs/2407.10672

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