Triple system

In algebra, a triple system (or ternar) is a vector space V over a field F together with a F-trilinear map

The decomposition of g is clearly a symmetric decomposition for this Lie bracket, and hence if G is a connected Lie group with Lie algebra g and K is a subgroup with Lie algebra k, then G/K is a symmetric space.

Conversely, given a Lie algebra g with such a symmetric decomposition (i.e., it is the Lie algebra of a symmetric space), the triple bracket [[u, v], w] makes m into a Lie triple system.

Any Jordan triple system is a Lie triple system with respect to the product

A Jordan triple system is said to be positive definite (resp. nondegenerate) if the bilinear form on V defined by the trace of Lu,v is positive definite (resp. nondegenerate). In either case, there is an identification of V with its dual space, and a corresponding involution on g0. They induce an involution of

which in the positive definite case is a Cartan involution. The corresponding symmetric space is a symmetric R-space. It has a noncompact dual given by replacing the Cartan involution by its composite with the involution equal to +1 on g0 and −1 on V and V*. 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).

A Jordan pair is a generalization of a Jordan triple system involving two vector spaces V+ and V. The trilinear map is then replaced by a pair of trilinear maps

which are often viewed as quadratic maps V+ → Hom(V, V+) and V → Hom(V+, V). The other Jordan axiom (apart from symmetry) is likewise replaced by two axioms, one being

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. The Jordan axioms (apart from symmetry) may then be written

which imply that the images of L+ and L are closed under commutator brackets in End(V+) and End(V). Together they determine a linear map

Jordan triple systems are Jordan pairs with V+ = V and equal trilinear maps. Another important case occurs when V+ and V are dual to one another, with dual trilinear maps determined by an element of