Structure of Alternative and Jordan Bimodules

Abstract

The notion of a bimodule for a class of algebras defined by multilinear identities has been introduced by Eilenberg [13]. If \( \mathfrak{A} \) is in the class of associative algebras or in the class of Lie algebras, then this notion is the familiar one for which we are in possession of well-worked theories. The study of bimodules (or representations) of Jordan algebras was initiated by the author in a recent paper [21]. Subsequently the alternative case was considered by Schafer [32]. In our paper we introduced the basic concepts of the Jordan theory and we proved complete reducibility of the bimodules and the analogue of Whitehead’s first lemma for finite dimensional semi-simple Jordan algebras of characteristic 0. Similar results on alternative algebras, based on those in the Jordan case, were obtained by Schafer. The principal tool in our paper was the notion of a Lie triple system. This permitted the application of important results on the structure and representation of Lie algebras to the problems on Jordan and alternative algebras. This method has one nice feature, namely, it is a general one which does not require a consideration of cases.