A structure theory of Lie triple systems

Abstract

If in an associative algebra 2I a new composition is introduced by putting [ab] = ab ba, 2I becomes a nonassociative algebra 2%(L with a skew-symmetric product satisfying the Jacobi identity [[ab]c]+ [[bc]a]+ [[ca]b] =0. Furthermore, any abstract algebra having these properties is isomorphic to a subalgebra of some IL. These algebras, called Lie algebras, have been extensively studied, and in this paper we shall often refer to some of the many known results. If, on the other hand, we consider the system denoted by Sij comprising 2I and the product {ab} =ab+ba, we observe that 2T is a commutative, nonassociative algebra such that { { { aa } b } a } = { { aa }, { ba } }. Such algebras and their subalgebras are special Jordan algebras. This terminology is necessary since they are not characterized by the above properties. Abstract algebras defined by these properties are simply Jordan algebras. The starting point for our discussion is the observation that in an associative algebra [[ab]c] = { {cb}a} { {ca}b}, so that any special Jordan algebra may be treated as a subspace of an associative algebra closed under triple Lie products. Jacobson has characterized subspaces of Lie algebras closed under triple Lie products and called them Lie triple systems. Further, if in any Jordan algebra 2I we introduce the ternary composition [abc] = { { cb } a } { { ca} b }, 2I becomes a Lie triple system, the associator Lie triple system of 2I. Also, the mappings xRa = { xa } in a Jordan algebra constitute a Lie triple system which is a homomorph of the associator Lie triple system and is called the multiplication Lie triple system of the Jordan algebra. Suggested by these relations is the possibility of using Lie algebra methods and results in the study of Jordan algebras. Jacobson [7](3) has successfully done this, and it is not our present purpose to pursue further such applications of Lie triple systems. We shall, rather, regard such connections as motivation for the abstract study of the structure and classification of these systems. In section I we introduce some fundamental concepts and note some results of Jacobson concerning imbeddings of Lie triple systems in Lie algebras. Section II develops notions of the radical, semi-simplicity, and solvability as defined for Lie triple systems including proofs of the existence of a semi-simple subsystem complementary to the radical and of the decomposi-