Groups of type E_7 over arbitrary fields

Details Groups of type E_7 over arbitrary fields Groups of type E_7 over arbitrary fields

1998-11-09

arxiv.org/abs/math/9811056v3

Communications in Algebra, vol. 29 (2001), 2689-2710

Mathematics

Algebraic Geometry

19 pages, AMS-LaTeX -- small errors corrected, last section revised

Scientific paper

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.

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