The Freudenthal-Springer-Tits constructions of exceptional Jordan algebras

Abstract

Constructions of 27-dimensional exceptional simple Jordan algebras have been given by H. Freudenthal [2], [3], T. A. Springer [8], and J. Tits [9]. In the first two approaches the cubic generic norm form plays a central role, with applications to projective geometry and algebraic groups; the third approach gives a simple method for constructing all exceptional simple algebras. The constructions are limited to fields of characteristic #2 as usual for Jordan algebras defined in terms of a bilinear multiplication, and in order to polarize the cubic norm form the characteristic must be # 3. Recently a definition of Jordan algebras has been proposed [5] which is based on a cubic composition involving U-operators. A unital Jordan algebra over a commutative associative ring &P is a triple : = (X, U, 1) where X is a &P-module, U a quadratic mapping x -> Ux of X into Homo (X, X), and 1 an element of X satisfying the axioms