Jordan triple system

Idea

Jordan algebras were invented to axiomatize some properties of the Jordan product a∘b=(ab+ba)/2a \circ b = (ab + ba)/2 of self-adjoint complex matrices. As work proceeded, some researchers found it convenient to focus on another binary operation on self-adjoint matrices, U a(b)=abaU_a (b) = a b a. Axiomatizing the properties of this led to the definition of a quadratic Jordan algebra?. However, it is often convenient to replace this operation U a(b)U_a(b) by a trilinear operation, by polarizing it as follows:

{a,b,c}=U a+c(b)−U a(b)−U c(b)=abc+cba {a,b,c} = U_{a+c}(b) - U_a(b) - U_…

Idea

{a,b,c}=U a+c(b)−U a(b)−U c(b)=abc+cba {a,b,c} = U_{a+c}(b) - U_a(b) - U_c(b) = a b c + c b a

A Jordan triple system axiomatizes the properties of this triple product of self-adjoint complex matrices.

Later it was discovered that there are close relations between Jordan triple systems and Lie triple systems. Just as Lie triple systems naturally give ℤ/2\mathbb{Z}/2-graded Lie algebras, Jordan triple systems naturally give a certain class of so-called 3-graded Lie algebras, which are ℤ\mathbb{Z}-graded Lie algebras concentrated in degrees −1,0-1,0 and 11. And just as the tangent space of any point in a symmetric space is naturally a Lie triple system, the tangent space of any point in a hermitian symmetric space is naturally a Jordan triple system.

Definition

A Jordan triple system is a vector space VV equipped with a trilinear map {⋅,⋯,⋅}:V×V×V→V{\cdot, \cdots, \cdot } \colon V \times V \times V \to V obeying two axioms:

{a,b,c}={c,b,a} {a,b,c} = {c,b,a}

{a,b,{c,d,e}}−{c,d,{a,b,e}}={{a,b,c},d,e}−{c,{b,a,d},e}{a,b,{c,d,e}} - {c,d,{a,b,e}} = {{a,b,c},d,e} - {c,{b,a,d},e}

Any subspace of an associative algebra closed under the operation {a,b,c}=abc+cba{a,b,c} = a b c + c b a obeys these axioms. The first axiom captures the symmetry of this operation under switching the first and last arguments, while the second, subtler axiom implies that the operations L a,b:V→VL_{a,b} \colon V \to V given by L a,b(c)={a,b,c}L_{a,b}(c) = {a,b,c} form a Lie algebra under commutators.