Triple system
# 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:

    $[u,v,w]=-[v,u,w]$
    $[u,v,w]+[w,u,v]+[v,w,u]=0$
    Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): [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 ${\mathfrak {g}}_{-1}$.

