Groups of type $\mathrm{E}_6$ and $\mathrm{E}_7$ over Rings via Brown Algebras and Related Torsors

TL;DR

This paper explores the structure of exceptional algebraic groups of types E6 and E7 over rings, using Brown algebras and torsors, and provides geometric descriptions of related homogeneous spaces.

Contribution

It introduces a new framework for realizing E6 and E7 groups over rings via Brown algebras and describes the parametrization of principal isotopes of Brown algebras.

Findings

Automorphism groups of Freudenthal triple systems are of type E7.

E6 groups are realized as centralizers within these structures.

Homogeneous spaces of E7/E6 parametrize principal isotopes of Brown algebras.

Abstract

We study structurable algebras and their associated Freudenthal triple systems over commutative rings. The automorphism groups of these triple systems are exceptional groups of type $\mathrm{E}_7$, and we realize groups of type $\mathrm{E}_6$ as centralizers. When 6 is invertible, we further give a geometric description of homogeneous spaces of type $\mathrm{E}_7/\mathrm{E}_6$, and show that they parametrize principal isotopes of Brown algebras. As opposed to the situation over fields, we show that such isotopes may be non-isomorphic.