Some structure theory for a class of triple systems

Abstract:

This paper deals with a class of triple systems satisfying two generalized five linear identities and having nondegenerate bilinear forms with certain properties. If $(M,\{ ,,\} )$ is such a triple system with bilinear form $\phi (,)$, it is shown that if $M$ is semisimple, then $M$ is the direct sum of simple ideals if $\phi$ is symmetric or symplectic or if $M$ is completely reducible as a module for its right multiplication algebra $\mathcal {L}$. It is also shown that if $M$ is a completely reducible $\mathcal {L}$-module, $M$ is the direct sum of a semisimple ideal and the center of $M$. Such triple systems can be embedded into certain nonassociative algebras and the results on the triple systems are extended to these algebras.