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

TL;DR

This paper investigates simple Lie algebras generated by extremal elements over arbitrary fields, establishing conditions under which they admit specific gradings linked to cubic norm and quadrangular algebras, including new exponential map definitions.

Contribution

It introduces new criteria for Lie algebras to admit $5 imes 5$-gradings parametrized by cubic norm and quadrangular algebras, extending understanding over arbitrary fields.

Findings

Lie algebras with extremal lines admit cubic norm structure gradings.

Existence of lines over field extensions and symplectic pairs leads to quadrangular algebra gradings.

New exponential map definitions work in characteristic 2 and 3.

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.