View source for Freudenthal-Kantor triple system

A triple system considered for constructing all simple Lie algebras (cf. [[Lie algebra|Lie algebra]]), and introduced as an algebraic system which is a generalization both of the algebraic systems appearing in the metasymplectic geometry developed by H. Freudenthal and of a generalized [[Jordan triple system|Jordan triple system]] of second order developed by I.L. Kantor.

Recall that a triple system is a [[Vector space|vector space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130240/f1302401.png" /> over a [[Field|field]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130240/f1302402.png" /> together with a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130240/f1302403.png" />-trilinear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130240/f1302404.png" />.

For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130240/f1302405.png" />, a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130240/f1302406.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130240/f1302407.png" /> with the trilinear product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130240/f1302408.png" /> is called a Freudenthal–Kantor 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/f/f130/f130240/f1302409.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/f/f130/f130240/f13024010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130240/f13024011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130240/f13024012.png" />.

In particular, a Freudenthal–Kantor triple system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130240/f13024013.png" /> is said to be balanced if there exists a bilinear form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130240/f13024014.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130240/f13024015.png" />, for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130240/f13024016.png" />.

This balancing property is closely related to metasymplectic geometry.

Note that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130240/f13024017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130240/f13024018.png" /> (identically), then the Freudenthal–Kantor triple system reduces to a Jordan triple system.

As the notion of a Freudenthal–Kantor triple system includes the notions of a generalized Jordan triple system of second order, a structurable algebra, and an [[Allison–Hein triple system|Allison–Hein triple system]], it is useful in obtaining all Lie algebras, without the use of root systems and Cartan matrices.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130240/f13024019.png" /> be a vector space with a bilinear form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130240/f13024020.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130240/f13024021.png" /> is a Freudenthal–Kantor triple system with respect to the triple product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130240/f13024022.png" />. In particular, it is important that the linear span <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130240/f13024023.png" /> of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130240/f13024024.png" /> makes a Jordan triple system of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130240/f13024025.png" /> with respect to the triple product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130240/f13024026.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130240/f13024027.png" /> be a Freudenthal–Kantor triple system. The vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130240/f13024028.png" /> becomes a [[Lie triple system|Lie triple system]] with respect to the triple product defined by

<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/f/f130/f130240/f13024029.png" /></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/f/f130/f130240/f13024030.png" /></td> </tr></table>

Using this, one can obtain the Lie triple system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130240/f13024031.png" /> associated with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130240/f13024032.png" />; it is denoted be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130240/f13024033.png" />.

Using the concept of the standard embedding Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130240/f13024034.png" /> associated with a Lie triple system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130240/f13024035.png" />, one can obtain the construction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130240/f13024036.png" /> associated with a Freudenthal–Kantor triple system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130240/f13024037.png" />. In fact, put

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130240/f13024038.png" /> equal to the linear span of the endomorphisms

<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/f/f130/f130240/f13024039.png" /></td> </tr></table>

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130240/f13024040.png" />;

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130240/f13024041.png" />;

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130240/f13024042.png" /> equal to the linear span of the endomorphisms

<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/f/f130/f130240/f13024043.png" /></td> </tr></table>

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130240/f13024044.png" /> equal to the linear span of the endomorphisms

<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/f/f130/f130240/f13024045.png" /></td> </tr></table>

Then one obtains the decomposition

<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/f/f130/f130240/f13024046.png" /></td> </tr></table>

and, more precisely,

<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/f/f130/f130240/f13024047.png" /></td> </tr></table>

These results imply the dimensional formula

<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/f/f130/f130240/f13024048.png" /></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/f/f130/f130240/f13024049.png" /></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/f/f130/f130240/f13024050.png" /></td> </tr></table>

This algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130240/f13024051.png" /> is called the Lie algebra associated with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130240/f13024052.png" />.

The concepts of a triple system and a supertriple system are important in the theory of quarks and Yang–Baxter equations.

Note that a  "triple system"  in the sense discussed above is totally different from  "triple system"  in combinatorics (see, e.g., [[Steiner system|Steiner system]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Freudenthal,   "Beziehungen der <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130240/f13024053.png" /> und <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130240/f13024054.png" /> zur Oktavenebene I–II"  ''Indag. Math.'' , '''16'''  (1954)  pp. 218–230; 363–386</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N. Kamiya,   "The construction of all simple Lie algebras over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130240/f13024055.png" /> from balanced Freudenthal–Kantor triple systems" , ''Contributions to General Algebra'' , '''7''' , Hölder–Pichler–Tempsky, Wien  (1991)  pp. 205–213</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  N. Kamiya,   "On Freudenthal–Kantor triple systems and generalized structurable algebras" , ''Non-Associative Algebra and Its Applications'' , Kluwer Acad. Publ.  (1994)  pp. 198–203</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  N. Kamiya,   S. Okubo,   "On <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130240/f13024056.png" />-Lie supertriple systems associated with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130240/f13024057.png" />-Freudenthal–Kantor supertriple systems"  ''Proc. Edinburgh Math. Soc.'' , '''43'''  (2000)  pp. 243–260</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  I.L. Kantor,   "Models of exceptional Lie algebras"  ''Soviet Math. Dokl.'' , '''14'''  (1973)  pp. 254–258</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  S. Okubo,   "Introduction to octonion and other non-associative algebras in physics" , Cambridge Univ. Press  (1995)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  K. Yamaguti,   "On the metasymplectic geometry and triple systems"  ''Surikaisekikenkyusho Kokyuroku, Res. Inst. Math. Sci. Kyoto Univ.'' , '''306'''  (1977)  pp. 55–92  (In Japanese)</TD></TR></table>