GROUPS OF TYPE E₇ OVER ARBITRARY FIELDS

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 ≠ 2, 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.

ACKNOWLEDGMENTS

I would like to thank an anonymous referee for their helpful suggestions.