# Triple system

In algebra, a triple system (or ternar) is a vector space V over a field F together with a F-trilinear map
    $(\cdot ,\cdot ,\cdot )\colon V\times V\times V\to V.$
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 [[u, v], w] and triple anticommutators {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 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 if the trilinear map, denoted $[\cdot ,\cdot ,\cdot ]$, satisfies the following identities:
    $[u,v,w]=-[v,u,w]$
    $[u,v,w]+[w,u,v]+[v,w,u]=0$
    $[u,v,[w,x,y]]=[[u,v,w],x,y]+[w,[u,v,x],y]+[w,x,[u,v,y]].$
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
    ${\mathfrak {g}}:=k\oplus {\mathfrak {m}}$
can be made into a $\mathbb {Z} _{2}$-graded Lie algebra, the standard embedding of m, with bracket
    $[(L,u),(M,v)]=([L,M]+L_{u,v},L(v)-M(u)).$
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:
    $\{u,v,w\}=\{u,w,v\}$
    $\{u,v,\{w,x,y\}\}=\{w,x,\{u,v,y\}\}+\{w,\{u,v,x\},y\}-\{\{v,u,w\},x,y\}.$
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
    $[L_{u,v},L_{w,x}]:=L_{u,v}\circ L_{w,x}-L_{w,x}\circ L_{u,v}=L_{w,\{u,v,x\}}-L_{\{v,u,w\},x}$
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 g$_0$.
Any Jordan triple system is a Lie triple system with respect to the product
    $[u,v,w]=\{u,v,w\}-\{v,u,w\}.$
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
    $V\oplus {\mathfrak {g}}_{0}\oplus V^{*}$
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).

## 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
    $\{\cdot ,\cdot ,\cdot \}_{+}\colon V_{-}\times S^{2}V_{+}\to V_{+}$
    $\{\cdot ,\cdot ,\cdot \}_{-}\colon V_{+}\times S^{2}V_{-}\to V_{-}$
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
    $\{u,v,\{w,x,y\}_{+}\}_{+}=\{w,x,\{u,v,y\}_{+}\}_{+}+\{w,\{u,v,x\}_{+},y\}_{+}-\{\{v,u,w\}_{-},x,y\}_{+}$
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
    $L_{u,v}^{+}:V_{+}\to V_{+}\quad {\text{by}}\quad L_{u,v}^{+}(y)=\{u,v,y\}_{+}$
and similarly L$^−$. The Jordan axioms (apart from symmetry) may then be written
    $[L_{u,v}^{\pm },L_{w,x}^{\pm }]=L_{w,\{u,v,x\}_{\pm }}^{\pm }-L_{\{v,u,w\}_{\mp },x}^{\pm }$
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
    $V_{+}\otimes V_{-}\to {\mathfrak {gl}}(V_{+})\oplus {\mathfrak {gl}}(V_{-})$
whose image is a Lie subalgebra ${\mathfrak {g}}_{0}$, and the Jordan identities become Jacobi identities for a graded Lie bracket on
    $V_{+}\oplus {\mathfrak {g}}_{0}\oplus V_{-},$
so that conversely, if
    ${\mathfrak {g}}={\mathfrak {g}}_{+1}\oplus {\mathfrak {g}}_{0}\oplus {\mathfrak {g}}_{-1}$
is a graded Lie algebra, then the pair $({\mathfrak {g}}_{+1},{\mathfrak {g}}_{-1})$ is a Jordan pair, with brackets
    $\{X_{\mp },Y_{\pm },Z_{\pm }\}_{\pm }:=[[X_{\mp },Y_{\pm }],Z_{\pm }].$
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
    $\mathrm {End} (S^{2}V_{+})\cong S^{2}V_{+}^{*}\otimes S^{2}V_{-}^{*}\cong \mathrm {End} (S^{2}V_{-}).$
These arise in particular when ${\mathfrak {g}}$ above is semisimple, when the Killing form provides a duality between ${\mathfrak {g}}_{+1}$ and ${\mathfrak {g}}_{-1}$.