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 most important examples are Lie triple systems and Jordan triple systems. They were introduced by Nathan Jacobson in 1949 to study subspaces of associative algebras closed under triple commutators anticommutator]]s {"u", {"v", "w"}}. 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 space]]s, particularly [[Hermitian symmetric space]]s and their generalizations ([[symmetric R-space]]s and their noncompact duals).
Lie triple systems.
A triple system is said to be a "Lie triple system" if the trilinear map, denoted formula_2, 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 formula_7-graded Lie algebra, the "standard embedding" of m, 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 map, 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 space]]s of compact and noncompact type (the latter being [[bounded symmetric domain]]s).
Jordan pair.
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
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 formula_20, 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 formula_23 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 formula_26 above is semisimple, when the Killing form provides a duality between formula_27 and formula_28.
References.
[[Category:Representation theory]]