View source for Jordan triple system

A triple system closely related to Jordan algebras.

A triple system is a [[Vector space|vector space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130060/j1300601.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130060/j1300602.png" /> together with a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130060/j1300603.png" />-trilinear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130060/j1300604.png" />, called a triple product and usually denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130060/j1300605.png" /> (sometimes dropping the commas).

It is said to be a Jordan triple system if

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130060/j1300606.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130060/j1300607.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130060/j1300608.png" /></td> </tr></table>

with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130060/j1300609.png" />.

From the algebraic viewpoint, a Jordan triple system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130060/j13006010.png" /> is a [[Lie triple system|Lie triple system]] with respect to the new triple product

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130060/j13006011.png" /></td> </tr></table>

This implies that all simple Lie algebras over an [[Algebraically closed field|algebraically closed field]] of characteristic zero, except <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130060/j13006012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130060/j13006013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130060/j13006014.png" /> (cf. also [[Lie algebra|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130060/j13006015.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130060/j13006016.png" /> be an associative algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130060/j13006017.png" /> (cf. also [[Associative rings and algebras|Associative rings and algebras]]) and set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130060/j13006018.png" />, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130060/j13006019.png" />-matrices over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130060/j13006020.png" />. This vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130060/j13006021.png" /> is a Jordan triple system with respect to the product

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130060/j13006022.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130060/j13006023.png" /> denotes the [[transpose matrix]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130060/j13006024.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130060/j13006025.png" /> be a vector space over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130060/j13006026.png" /> equipped with a symmetric bilinear form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130060/j13006027.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130060/j13006028.png" /> is a Jordan triple system with respect to the product

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130060/j13006029.png" /></td> </tr></table>

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130060/j13006030.png" /> be a commutative [[Jordan algebra|Jordan algebra]]. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130060/j13006031.png" /> is a Jordan triple system with respect to the product

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130060/j13006032.png" /></td> </tr></table>

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, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130060/j13006033.png" />-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. Nehr,   "Jordan triple systems by the graid approach" , ''Lecture Notes in Mathematics'' , '''1280''' , Springer  (1987)</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>