A theorem on the derivations of Jordan algebras

Abstract

The restriction on the dimensionality of the simple components arises from the fact that the (3-dimensional) central simple Jordan algebra of all 2 X 2 symmetric matrices has for its derivation algebra the abelian Lie algebra of dimension 1. However, most simple Jordan algebras over F have simple derivation algebras, and all except those of dimension 3 over their centers have derivation algebras which are semisimple or {0 }, as may be seen from the listing by N. Jacobson [3, ?4] of these derivation algebras.2 The part of the theorem then follows from the direct sum relationship. To demonstrate the converse it is sufficient to show that, if Z is semisimple or {0}, then 2( is semisimple. For then, if any simple component of 2t had dimension 3 over its center, it would have an abelian derivation algebra not equal to {0 } [3, ?4], which would give rise to a nonzero abelian ideal in Z, a contradiction. To show that 2t is semisimple whenever Z is semisimple or {0}, we use the so-called Wedderburn Principal Theorem for Jordan algebras, proved recently by A. J. Penico [5]. Also Lemma 1 below is taken from the proof of that theorem [5, ?2]. Revisions have been made in the proof of our theorem in accordance with helpful suggestions of Professor Jacobson.