Triple system

In algebra, a triple system 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. 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.

Lie triple systems

A triple system is said to be a Lie triple system if the trilinear map, denoted, satisfies the following identities:
The first two identities abstract the skew symmetry and Jacobi identity for the triple commutator, while the third identity means that the linear map L_u,''v: V'' → V, defined by L_u,''v =, is a derivation of the triple product. The identity also shows that the space of linear operators = span is closed under commutator bracket, hence a Lie algebra.
It follows that
is a -graded Lie algebra with of grade 0 and V'' of grade 1, and bracket
This is called the standard embedding of the Lie triple system V into a -graded Lie algebra. Conversely, given any -graded Lie algebra, the triple bracket u, v], w] makes the space of degree-1 elements into a Lie triple system.
However, these methods of converting a Lie triple system into a -graded Lie algebra and vice versa are not inverses: more precisely, they do not define an equivalence of categories. For example, if we start with any abelian -graded Lie algebra, the round trip process produces one where the grade-0 space is zero-dimensional, since we obtain = span =.
Given any Lie triple system V, and letting V be the corresponding -graded Lie algebra, this decomposition of obeys the algebraic definition of a symmetric space, so if G is any connected Lie group with Lie algebra and H is a subgroup with Lie algebra, then G/''H is a symmetric space. Conversely, the tangent space of any point in any symmetric space is naturally a Lie triple system.
We can also obtain Lie triple systems from associative algebras. Given an associative algebra A'' and defining the commutator by, any subspace of A closed under the operation
becomes a Lie triple system with this operation.

Jordan triple systems

A triple system V is said to be a Jordan triple system if the trilinear map, denoted, satisfies the following identities:
The second identity means that if L_u,''v:V''→V is defined by L_u,''v = then
so that the space of linear maps span is closed under commutator bracket, and hence is a Lie algebra.
A Jordan triple system is said to be positive definite if the bilinear form on V'' defined by the trace of L_u,''v is positive definite. In either case, there is an identification of V'' with its dual space, and a corresponding involution on. 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 and −1 on V and V^*. A special case of this construction arises when preserves a complex structure on V. In this case we obtain dual Hermitian symmetric spaces of compact and noncompact type.
Any Jordan triple system is a Lie triple system with respect to the operation

Jordan pairs

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
The other Jordan axiom is likewise replaced by two axioms, one being
and the other being the analogue with + and − subscripts exchanged. The trilinear maps are often viewed as quadratic maps
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 may then be written
which imply that the images of L⁺ and L⁻ are closed under commutator brackets in End and End. 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
making this space into a -graded Lie algebra with only grades 1, 0, and -1 being nontrivial, often called a 3-graded Lie algebra. Conversely, given any 3-graded Lie algebra
then the pair is a Jordan pair, with brackets
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
These arise in particular when above is semisimple, when the Killing form provides a duality between and.
For a simple example of a Jordan pair, let be a finite-dimensional vector space and the dual of that vector space, with the quadratic maps
given by
where.