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 , { , , } ) (M,\{ ,,\} ) is such a triple system with bilinear form ϕ ( , ) \phi (,) , it is shown that if M M is semisimple, then M M is the direct sum of simple ideals if ϕ \phi is symmetric or symplectic or if M M is completely reducible as a module for its right multiplication algebra L \mathcal {L} . It is also shown that if M M is a completely reducible L \mathcal {L} -module, M M is the direct sum of a semisimple ideal and the center of M M . Such triple systems can be embedded into certain nonassociative algebras and the results on the triple systems are extended to these algebras.