Jordan triple system

A triple system closely related to Jordan algebras.

A triple system is a vector space over a field together with a -trilinear mapping , called a triple product and usually denoted by (sometimes dropping the commas).

It is said to be a Jordan triple system if

(a1)

(a2)

with .

From the algebraic viewpoint, a Jordan triple system is a Lie triple system with respect to the new triple product

This implies that all simple Lie algebras over an algebraically closed field of characteristic zero, except , and (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 -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 be an associative algebra over (cf. also Associative rings and algebras) and set , the -matrices over . This vector space is a Jordan triple system with respect to the product

where denotes the transpose matrix of .

Let be a vector space over equipped with a symmetric bilinear form . Then is a Jordan triple system with respect to the product

Let be a commutative Jordan algebra. Then is a Jordan triple system with respect to the product

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, -symmetric spaces, and bounded symmetric domains" Bull. Amer. Math. Soc. , 77 (1971) pp. 558–561
[a4]	E. Nehr, "Jordan triple systems by the graid approach" , Lecture Notes in Mathematics , 1280 , Springer (1987)
[a5]	S. Okubo, N. Kamiya, "Jordan–Lie super algebra and Jordan–Lie triple system" J. Algebra , 198 : 2 (1997) pp. 388–411