View source for Jordan triple system

A triple system closely related to [[Jordan algebra]]s.

A triple system is a [[vector space]] $V$ over a field $K$ together with a $K$-[[trilinear mapping]] $V \times V \times V \rightarrow V$, called a ''triple product'' and usually denoted by $\{ \cdot , \cdot , \cdot \}$ (sometimes dropping the commas).

It is said to be a Jordan triple system if
$$
\{ u,v,w \} = \{ w,v,u \} \ ,
$$
$$
\{x,y,\{u,v,w\}\} = \{\{x,y,u\},v,w\} - \{u,\{y,x,v\},w\} + \{u,v,\{x,y,w\}\}
$$
with $x,y,u,v,w \in V$.

From the algebraic viewpoint, a Jordan triple system $(V,\{,,\})$ is a [[Lie triple system]] with respect to the new triple product
$$
[x,y,z] = \{x,y,z\} - \{y,x,z\} \ .
$$

This implies that all simple Lie algebras over an [[algebraically closed field]] of characteristic zero, except $G_2$, $F_4$ and $E_8$ (cf. also [[Lie algebra]]), can be constructed using the standard embedding Lie algebra associated with a Lie triple system via a Lie triple system.

From the geometrical viewpoint there is, for example, a correspondence between symmetric $R$-spaces and compact Jordan triple systems [[#References|[a3]]] as well as a correspondence between bounded symmetric domains and Hermitian Jordan triple systems [[#References|[a2]]].

For superversions of this triple system, see [[#References|[a5]]].

==Examples.==
Let $D$ be an associative algebra over $K$ (cf. also [[Associative rings and algebras]]) and set $V = M_{p,q}(D)$, the $(p\times q)$-matrices over $D$. This vector space $V$ is a Jordan triple system with respect to the product
$$
\{x,y,z\} = x y^\top z + z y^\top x
$$
where $y^\top$ denotes the [[transpose matrix]] of $y$.

Let $V$ be a vector space over $K$ equipped with a symmetric bilinear form $(\cdot,\cdot)$. Then $V$ is a Jordan triple system with respect to the product
$$
\{x,y,z\} = (x,y) z + (y,z) x - y (z,x) \ .
$$

Let $J$ be a commutative [[Jordan algebra]]. Then $J$ is a Jordan triple system with respect to the product
$$
\{x,y,z\} = x(yz) + (xy)z - y(xz) \ .
$$

Note that a triple system in this sense is completely different from, e.g., the combinatorial notion of a Steiner triple system (cf. also [[Steiner system|Steiner system]]).

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Jacobson,   "Lie and Jordan triple systems"  ''Amer. J. Math.'' , '''71'''  (1949)  pp. 149–170</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Kaup,   "Hermitian Jordan triple systems and the automorphisms of bounded symmetric domains" , ''Non Associative Algebra and Its Applications (Oviedo, 1993)'' , Kluwer Acad. Publ.  (1994)  pp. 204–214</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  O. Loos,   "Jordan triple systems, $R$-symmetric spaces, and bounded symmetric domains"  ''Bull. Amer. Math. Soc.'' , '''77'''  (1971)  pp. 558–561</TD></TR>
<TR><TD valign="top">[a4]</TD> <TD valign="top">  E. Neher,   "Jordan triple systems by the grid approach" , ''Lecture Notes in Mathematics'' , '''1280''' , Springer  (1987)  {{ZBL|0621.17001}}</TD></TR>
<TR><TD valign="top">[a5]</TD> <TD valign="top">  S. Okubo,   N. Kamiya,   "Jordan–Lie super algebra and Jordan–Lie triple system"  ''J. Algebra'' , '''198''' :  2  (1997)  pp. 388–411</TD></TR>
</table>

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