Lie and Jordan Triple Systems

Abstract

The present paper is devoted to a study of subspaces of an associative algebra that are closed relative to the ternary operation [[a, b], c] where \( \left[ {a,b} \right] = ab - ba \) Such systems—called Lie triple systems—arise in a natural way in the study of Jordan algebras and of Jordan triple systems. The latter are defined to be subspaces of an associative algebra that are closed relative to {{a, b}, c} where \( \left\{ {a,b} \right\} = ab + ba \). In the first part of this paper we consider some general properties of such systems. The second half of our paper is concerned with the study of certain particular Lie and Jordan triple systems that have arisen in quantum mechanics. These systems have a basis g 1 g 2,…, g n and multiplication tables, respectively $$ \begin{gathered} \left[ {\left[ {g_i ,g_j } \right],g_k } \right] = \delta _{ki} g_j - \delta _{kj} g_i \hfill g_i g_j g_k + g_k g_j g_i = - \delta _{ij} g_k - \delta _{kj} g_i \hfill \end{gathered} $$ The latter relations have been introduced by Duffin 1 and by Kemmer 2 in the study of meson fields and there is an extensive literature on the representation theory of such systems. In this paper we consider an extension of this theory.