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

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

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 form, denoted [.,.,.], 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

$\mathfrak g := \mathfrak k \oplus\mathfrak m$

can be made into a Lie algebra 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 form, 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 form is then replaced by a pair of trilinear forms

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

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

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

$[L^{\pm}_{u,v},L^{\pm}_{w,x}] = L^{\pm}_{w,\{u,v,x\}_\pm}-L^{\pm}_{\{v,u,w\}_{\mp},x}$

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

$\mathrm{End}(S^2V_+) \cong S^2V_+^* \otimes S^2V_-^*\cong \mathrm{End}(S^2V_-).$

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

• Quadratic Jordan algebra

References

• (1st edition: Academic Press, New York, 1978).
• .
• .
• doi:10.1090/S0002-9904-1971-12753-2