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, ![]() |
[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 |
Jordan triple system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Jordan_triple_system&oldid=11502