The primary aim of this paper is to study mappings J of rings that are additive and that satisfy the conditions 

$$ {\left( {{a^2}} \right)^J} = {\left( {{a^J}} \right)^2},\;{\left( {aba} \right)^J} = {a^J}{b^J}{a^J} $$

(1)

Such mappings will be called Jordan homomorphisms. If the additive groups admit the operator 1/2 in the sense that 2x = a has a unique solution (1/2)a for every a, then conditions (1) are equivalent to the simpler condition 

$$ {\left( {ab} \right)^J} + {\left( {ba} \right)^J} = {a^J}{b^J} + {b^J}{a^J} $$

(2)

Mappings satisfying (2) were first considered by Ancochea [1], [2](1). The modification to (1) is essentially due to Kaplansky [13]. Its purpose is to obviate the necessity of imposing any restriction on the additive groups of the rings under consideration.

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Jordan homomorphisms of rings

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Lie and Jordan structures in simple associative rings

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.

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Structure of Alternative and Jordan Bimodules

It is well known that the Lie algebra of derivations of the exceptional Jordan algebra M 3 8 over an algebraically closed field of characteristic 0 or over the field of reals is the exceptional Lie algebra F 4 of Killing-Cartan and that the Lie algebra of linear transformations in M 3 8 leaving a certain cubic norm form defined on M 3 8 invariant is the exceptional Lie algebra E 6 (Chevalley-Schafer [7], Freudenthal [11]). The program envisaged in the present series of papers is the extension of these results and their group analogues to arbitrary central simple Jordan algebras. The present paper gives the definition of the groups and Lie algebras in the general case and gives their determination for special Jordan algebras with some restrictions on the base field. In a second paper we plan to consider the groups of automorphisms of arbitrary reduced exceptional simple Jordan algebras and in a third paper we hope to study the generalized groups of types E 6 which arise from these Jordan algebras.

German Ancohea

Papers

On Semi-Automorphisms of Division Algebras