From cubic norm pairs to $G_2$- and $F_4$-graded groups and Lie algebras

TL;DR

This paper develops a framework for constructing Lie algebras and associated groups from cubic norm pairs, revealing new gradings by root systems of types G2 and F4, and providing detailed structural insights.

Contribution

It introduces a method to build Lie algebras from cubic norm pairs over arbitrary rings, with gradings by G2 and F4, and constructs corresponding root graded groups.

Findings

Lie algebras admit G2 gradings over arbitrary rings

F4-gradings are obtained when the pair is a cubic Jordan algebra

Detailed structural properties of these Lie algebras and groups are established

Abstract

We construct Lie algebras arising from cubic norm pairs over arbitrary commutative base rings. Such Lie algebras admit a grading by a root system of type $G_2$, and when the cubic norm pair is a cubic Jordan matrix algebra, the $G_2$-grading can be further refined to an $F_4$-grading. We then use these Lie algebras and their gradings to construct corresponding root graded groups. Along the way, we produce many results providing detailed information about the structure of these Lie algebras and groups.