Hermitian Cubic norm structures and groups of relative rank one

TL;DR

This paper generalizes the quartic norm for Hermitian cubic norm structures, classifies division structures via Tits indices, and links these structures to adjoint simple linear algebraic groups of relative rank one.

Contribution

It introduces the notion of division Hermitian cubic norm structures and classifies them using Tits indices, connecting algebraic structures with linear algebraic groups.

Findings

Elements with invertible quartic norm are conjugate invertible.

Classification of division structures via Tits index.

Correspondence between algebraic groups and hermitian cubic norm structures.

Abstract

Hermitian cubic norm structures were recently introduced in order to study the class of skew-dimension one structurable algebras (which are typically only defined over fields of characteristic different from $2$ and $3$) over arbitrary rings and fields. Here, we generalize the quartic norm for these algebras and show that elements for which the quartic norm is invertible are conjugate invertible. This leads to the notion of a division hermitian cubic norm structure, defined as one in which the quartic norm is anisotropic. We classify such structures in terms of the Tits index of an associated (rank one) adjoint simple linear algebraic group and show that any adjoint linear algebraic group with such a Tits index defines a corresponding hermitian cubic norm structure.