Groups of type E_7 over arbitrary fields

TL;DR

This paper introduces new algebraic structures that characterize all adjoint algebraic groups of type E_7 over arbitrary fields, incorporating a novel invariant and providing a comprehensive understanding of their properties.

Contribution

It defines an identity for nondegenerate Freudenthal triple systems and constructs algebraic structures that produce all E_7 groups, including a new invariant involving symplectic involutions.

Findings

Characterization of nondegenerate triple systems via a new identity

Construction of algebraic structures for all E_7 groups over arbitrary fields

Complete description of involutions in E_7 groups with Tits algebras of index 2

Abstract

Freudenthal triple systems come in two flavors, degenerate and nondegenerate. The best criterion for distinguishing between the two which is available in the literature is by descent. We provide an identity which is satisfied only by nondegenerate triple systems. We then use this to define algebraic structures whose automorphism groups produce all adjoint algebraic groups of type E_7 over an arbitrary field of characteristic not 2 or 3. The main advantage of these new structures is that they incorporate a previously unconsidered invariant (a symplectic involution) of these groups in a fundamental way. As an application, we give a construction of adjoint groups with Tits algebras of index 2 which provides a complete description of this involution and apply this to groups of type E_7 over a real-closed field.