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. 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).

Lie triple systems

A triple system is said to be a Lie triple system, 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 Lu,v:V→V, defined by Lu,v(w) = [u, v, w], is a derivation = span {Lu,v : u, v ∈ V} is closed under commutator bracket, hence a Lie algebra.

It follows that

-graded Lie algebra of grade 0 and V of grade 1, and bracket

This is called the standard embedding of the Lie triple system V-graded Lie algebra, the triple bracket [[u, v], w] makes the space of degree-1 elements into a Lie triple system.

-graded Lie algebra and vice versa are not inverses: more precisely, they do not define an equivalence of categories = span {Lu,v : u, v ∈ V} = {0}.

Given any Lie triple system V obeys the algebraic definition of a symmetric space, so if G is any connected Lie group and H, 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 any Lie algebra by setting

Jordan triple systems

A triple system V is said to be a Jordan triple system, satisfies the following identities:

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.

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. They induce an involution of

which in the positive definite case is a Cartan involution. The corresponding symmetric space is a symmetric R-space and 1 on V and V* 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).

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

1. .

The other Jordan axiom (apart from symmetry) 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 (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

, and the Jordan identities become Jacobi identities for a graded Lie bracket on

-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

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

above is semisimple, when the Killing form.

the dual of that vector space, with the quadratic maps

given by

.

See also

• Associator

• E7 (mathematics)

• Quadratic Jordan algebra

References

• .

• .