Triple system

For an account of that concept in combinatorics, see Steiner triple system and block design.

In algebra, a triple system (or ternar) is a vector space V over a field F together with a F-trilinear map

$(\cdot ,\cdot ,\cdot ):V\times V\times V\to V.$

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 ${\mathbb{\mathbb{Z}}}_2$-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 g₀. 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 g₀. 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 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 map is then replaced by a pair of trilinear maps

$\{\cdot ,\cdot ,\cdot {\}}_{+}:V_{-}\times S^2{V}_{+}\to V_{+}$

$\{\cdot ,\cdot ,\cdot {\}}_{-}:V_{+}\times S^2{V}_{-}\to V_{-}$

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

$L_{u,v}^{+}:V_{+}\to V_{+}\text{by}{L}_{u,v}^{+}(y)=\{u,v,y{\}}_{+}$

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

$V_{+}\otimes V_{-}\to \mathfrak{𝔤}\mathfrak{𝔩}(V_{+})\oplus \mathfrak{𝔤}\mathfrak{𝔩}(V_{-})$

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

$\mathrm{E}\mathrm{n}\mathrm{d}(S^2{V}_{+})\cong S^2{V}_{+}^{*}\otimes S^2{V}_{-}^{*}\cong \mathrm{E}\mathrm{n}\mathrm{d}(S^2{V}_{-}).$

These arise in particular when $\mathfrak{𝔤}$ above is semisimple, when the Killing form provides a duality between ${\mathfrak{𝔤}}_{+1}$ and ${\mathfrak{𝔤}}_{-1}$.

See also

• Associator
• Quadratic Jordan algebra

References

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