A Construction of Exceptional Jordan Division Algebras

Abstract

Every simple Jordan algebra ,) over a field H of characteristic not two has a unity element 1. Write 1 = e, + *-+ et for pairwise orthogonal primitive idempotents e,, and let 4? be the set of all elements xf of 1? such that em xi xi, where ab is the product in '?. Then g is said to be reduced of degree t if the t subalgebras As all have dimension one over A. If W is any associative algebra, and ab is the associative product of a, there is an attached Jordan algebra A'+) which consists of the vector space 2 and the product ab defined by 2ab = ab + ba. A Jordan algebra ? is said to be exceptional if ? is not isomorphic to a subalgebra of any W for W associative. It is known2 that the dimension of any simple exceptional Jordan algebra 1? over its center g is 27, and that there exists a scalar extension A, of finite degree over A, such that & is isomorphic to the Jordan algebra &3((S) of all three-rowed Hermitian matrices with elements in the split Cayley algebra (E of dimension 8 over R. Represent the product operation in the algebra (2, of all three-rowed matrices with elements in (E by ab, and the operation ab in 5?3(() is defined by 2ab ab + ba. In 1948 R. D. Schafer stated3 that every exceptional simple Jordan algebra is reduced. His proof of the statement contains an error4, and the result is false. We shall construct a class of central simple exceptional Jordan algebras ? over any field g of cearacteristic not two which we shall call cyclic Jordan algebras. Each such algebra & has an attached cyclic associative algebra = (Sty Sy r) of degree three over its center A, and an attached Cayley algebra (E over St having a nonsingular linear transformation T over g inducing S in St and such that