5-Graded simple Lie algebras, structurable algebras, and Kantor pairs

TL;DR

This paper classifies finite-dimensional 5-graded simple Lie algebras over fields with characteristic not 2 or 3, showing they are of Chevalley type, and connects them to structurable algebras and Kantor pairs.

Contribution

It establishes that all such 5-graded Lie algebras are of Chevalley type and links them to structurable algebras and Kantor pairs.

Findings

All finite-dimensional central simple 5-graded Lie algebras are of Chevalley type.

Every structurable algebra and Kantor pair over the field arises from a 5-grading of a Chevalley-type Lie algebra.

The classification relies on the known classification of simple Lie algebras over algebraically closed fields.

Abstract

Relying on the classification of simple Lie algebras over algebraically closed fields of characteristic $>3$, we show that any finite-dimensional central simple 5-graded Lie algebra over a field $k$ of characteristic $\neq 2,3$ is a simple Lie algebra of Chevalley type, i.e. a central quotient of the Lie algebra of a simple algebraic $k$-group. As a consequence, we prove that all central simple structurable algebras and Kantor pairs over $k$ arise from 5-gradings on simple Lie algebras of Chevalley type.