# 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 L_{u,v} is positive definite (resp. nondegenerate). In either case, there is an identification of *V* with its dual space, and a corresponding involution on **g**_{0}. 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 **g**_{0} and −1 on *V* and *V*^{*}. A special case of this construction arises when **g**_{0} 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