Triple system

In algebra, a triple system (or ternar) is a vector space V over a field F together with a F-trilinear map In algebra, a triple system (or ternar) is a vector space V over a field F together with a F-trilinear map The most important examples are Lie triple systems and Jordan triple systems. They were introduced by Nathan Jacobson in 1949 to study subspaces of associative algebras closed under triple commutators , w] and triple anticommutators {u, {v, w}}. In particular, any Lie algebra defines a Lie triple system and any Jordan algebra defines a Jordan triple system. They are important in the theories of symmetric spaces, particularly Hermitian symmetric spaces and their generalizations (symmetric R-spaces and their noncompact duals).