Jordan triple system

A triple system closely related to Jordan algebras.

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 [a3] as well as a correspondence between bounded symmetric domains and Hermitian Jordan triple systems [a2].

For superversions of this triple system, see [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).

References

[a1]	N. Jacobson, "Lie and Jordan triple systems" Amer. J. Math. , 71 (1949) pp. 149–170
[a2]	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
[a3]	O. Loos, "Jordan triple systems, $R$-symmetric spaces, and bounded symmetric domains" Bull. Amer. Math. Soc. , 77 (1971) pp. 558–561
[a4]	E. Neher, "Jordan triple systems by the grid approach" , Lecture Notes in Mathematics , 1280 , Springer (1987) Zbl 0621.17001
[a5]	S. Okubo, N. Kamiya, "Jordan–Lie super algebra and Jordan–Lie triple system" J. Algebra , 198 : 2 (1997) pp. 388–411