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 [[u, v], 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.
The identity also shows that the space k = span {Lu,v : u, v ∈ V} is closed under commutator bracket, hence a Lie algebra.
}, satisfies the following identities: The first identity abstracts the symmetry of the triple anticommutator, while the second identity means that if Lu,v:V→V is defined by Lu,v(y) = {u, v, y} then so that the space of linear maps span {Lu,v:u,v ∈ V} is closed under commutator bracket, and hence is a Lie algebra g0.
nondegenerate) if the bilinear form on V defined by the trace of Lu,v is positive definite (resp.
In either case, there is an identification of V with its dual space, and a corresponding involution on g0.
A special case of this construction arises when g0 preserves a complex structure on V. In this case we obtain dual Hermitian symmetric spaces of compact and noncompact type (the latter being bounded symmetric domains).
As in the case of Jordan triple systems, one can define, for u in V− and v in V+, a linear map and similarly L−.
Together they determine a linear map whose image is a Lie subalgebra
, and the Jordan identities become Jacobi identities for a graded Lie bracket on so that conversely, if is a graded Lie algebra, then the pair