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Difference between revisions of "Jordan triple system"

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A triple system closely related to [[Jordan algebra]]s.
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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).
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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
 
It is said to be a Jordan triple system if
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$$
 
$$
  
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.
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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 Jordan 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]]].
 
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]]].
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$$
 
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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]]).
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Note that a triple system in this sense is completely different from, e.g., the combinatorial notion of a [[Steiner triple system]].
  
 
====References====
 
====References====

Latest revision as of 10:13, 30 April 2018

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 Jordan 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.

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
How to Cite This Entry:
Jordan triple system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Jordan_triple_system&oldid=41895
This article was adapted from an original article by Noriaki Kamiya (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article