On special Jordan algebras

Abstract

for a and b in e is called quasimultiplication, and any linear subspace X over 8 of e which is closed with respect to this operation forms a corresponding algebra W. We call an algebra isomorphic to such an algebra a special Jordan algebra and see that special Jordan algebras are commutative but not, in general, associative. In the first three of four papers [6-9](1) Jordan considered the class of all algebras satisfying the property x(yx2) = (xy)x2 for all x and y. In the last paper the assumptions of reality and commutativity are made. It is the purpose of this paper to begin a study of simple special Jordan algebras. Any associative algebra e forms a special Jordan algebra (0 with respect to quasimultiplication and we shall show that if e is simple so is SO. If e has an involution J and SJ is the set of all J-symmetric quantities of S, the set SJ is a special Jordan algebra under quasimultiplication. Then we shall show again that es is simple if e is. Our study is analogous to that which has already been made for Lie algebras by Jacobson and Landherr [3, 4, 5, 10, It], and we shall show, as in that theory, that if 9 is a scalar extension of a such that Wq is a special Jordan algebra of one of the types above, then W is a special Jordan algebra of a corresponding type. The analogous analysis for Lie algebras formed a major part of the determination of all simple Lie algebras over any field of characteristic not two and it is expected that our study will occupy a corresponding place in the theory of Jordan algebras. Many thanks are due to Professor A. A. Albert for suggesting the topic, and guiding the writing, of the present paper which was accepted as a doctoral thesis at the University of Chicago. 2. Involutions. An involution of an associative algebra (E over W is a linear transformation J over a of e such that J2 =1, the identity transformation, (ab)J= bJaJ for every a and b of S. The quantities a = aJ of e are called J-symmetric and span a linear subspace 25j of E. The quantities a= -aJ of are called J-skew and span a linear subspace Es of S such that S is the