Triple system

Lie triple systems

A triple system is said to be a Lie triple system if the trilinear form, 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(w) = [u, v, w], is a derivation of the triple product. The identity also shows that the space k = span {L_u,v: u, v ∈ V} is closed under commutator bracket, hence a Lie algebra.

Writing m in place of V, it follows that

can be made into a Lie algebra with bracket

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.

Jordan triple systems

A triple system is said to be a Jordan triple system if the trilinear form, denoted {.,.,.}, satisfies the following identities:

The first identity abstracts the symmetry of the triple anticommutator, while the second identity means that if L_u,v:V→V is defined by L_u,v(y) = {u, v, y} then

so that the space of linear maps span {L_u,v:u,v ∈ V} is closed under commutator bracket, and hence is a Lie algebra g₀.

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 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).

Jordan pair

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

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

and the other being the analogue with + and − subscripts exchanged.

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

whose image is a Lie subalgebra g 0 , 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 ( g + 1 , g − 1 ) is a Jordan pair, with brackets

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

These arise in particular when g above is semisimple, when the Killing form provides a duality between g + 1 and g − 1 .