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The concept of a triple system, i.e. a [[Vector space|vector space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130200/a1302001.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130200/a1302002.png" /> together with a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130200/a1302003.png" />-[[trilinear mapping]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130200/a1302004.png" />, is mainly used in the theory of non-associative algebras and appears in the construction of Lie algebras (cf. also [[Lie algebra|Lie algebra]]; [[Non-associative rings and algebras|Non-associative rings and algebras]]).
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<!--This article has been texified automatically. Since there was no Nroff source code for this article,
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A [[Module|module]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130200/a1302005.png" /> over a field of characteristic not equal to two or three together with a trilinear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130200/a1302006.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130200/a1302007.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130200/a1302008.png" /> is said to be an Allison–Hein triple system (or a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130200/a13020010.png" />-ternary algebra) if
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<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/a/a130/a130200/a13020011.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
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The concept of a [[triple system]], i.e. a [[Vector space|vector space]] $V$ over a field $K$ together with a $K$-[[trilinear mapping]] $V \times V \times V \rightarrow V$, is mainly used in the theory of non-associative algebras and appears in the construction of Lie algebras (cf. also [[Lie algebra|Lie algebra]]; [[Non-associative rings and algebras|Non-associative rings and algebras]]).
  
<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/a/a130/a130200/a13020012.png" /></td> </tr></table>
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A [[Module|module]] $V$ over a field of characteristic not equal to two or three together with a trilinear mapping $( x , y , z ) \mapsto \langle x y z \rangle$ from $V \times V \times V$ to $V$ is said to be an Allison–Hein triple system (or a $J$-ternary algebra) 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/a/a130/a130200/a13020013.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
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\begin{equation} \tag{a1} \langle x y \langle u v w \rangle \rangle =  
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\langle \langle x y u \rangle v w \rangle + \langle u \langle y x v \rangle w \rangle + \langle u v \langle x y w \rangle \rangle
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\end{equation}
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130200/a13020014.png" />.
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\begin{equation} \tag{a2} \langle x y z \rangle - \langle z y x \rangle = \langle z x y \rangle - \langle x z y \rangle \end{equation}
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for all $x , y , z , u , v , w \in V$.
  
 
From the identities (a1) and (a2) one deduces the relation
 
From the identities (a1) and (a2) one deduces the relation
  
<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/a/a130/a130200/a13020015.png" /></td> </tr></table>
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\begin{equation*} K ( \langle a b c ) , d ) + K ( c , \langle a b d \rangle ) + K ( a , K ( c , d ) b ) = 0, \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130200/a13020016.png" />. Hence this triple system may be regarded as a variation of a [[Freudenthal–Kantor triple system|Freudenthal–Kantor triple system]]. In particular, it is important that the linear span <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130200/a13020017.png" /> of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130200/a13020018.png" /> is a Jordan subalgebra (cf. also [[Jordan algebra|Jordan algebra]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130200/a13020019.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130200/a13020020.png" />.
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where $K ( a , b ) c = \langle a c b \rangle - \langle b c a \rangle$. Hence this triple system may be regarded as a variation of a [[Freudenthal–Kantor triple system]]. In particular, it is important that the [[linear span]] $\{ K ( a , b ) \} _ { \operatorname{span} }$ of the set $K ( a , b )$ is a Jordan subalgebra (cf. also [[Jordan algebra]]) of $( \text { End } V ) ^ { + }$ with respect to $A \circ B = ( A B + B A ) / 2$.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B.N. Allison,  "A construction of Lie algebras from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130200/a13020021.png" />-ternary algebras"  ''Amer. J. Math.'' , '''98'''  (1976)  pp. 285–294</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Hein,  "A construction of Lie algebras by triple systems"  ''Trans. Amer. Math. Soc.'' , '''205'''  (1975)  pp. 79–95</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  N. Kamiya,  "A structure theory of Freudenthal–Kantor triple systems II"  ''Commun. Math. Univ. Sancti Pauli'' , '''38'''  (1989)  pp. 41–60</TD></TR><TR><TD valign="top">[a4]</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>
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<table>
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<tr><td valign="top">[a1]</td> <td valign="top">  B.N. Allison,  "A construction of Lie algebras from $J$-ternary algebras"  ''Amer. J. Math.'' , '''98'''  (1976)  pp. 285–294</td></tr>
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<tr><td valign="top">[a2]</td> <td valign="top">  W. Hein,  "A construction of Lie algebras by triple systems"  ''Trans. Amer. Math. Soc.'' , '''205'''  (1975)  pp. 79–95</td></tr>
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<tr><td valign="top">[a3]</td> <td valign="top">  N. Kamiya,  "A structure theory of Freudenthal–Kantor triple systems II"  ''Commun. Math. Univ. Sancti Pauli'' , '''38'''  (1989)  pp. 41–60</td></tr>
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<tr><td valign="top">[a4]</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>
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</table>

Latest revision as of 19:51, 27 February 2021

The concept of a triple system, i.e. a vector space $V$ over a field $K$ together with a $K$-trilinear mapping $V \times V \times V \rightarrow V$, is mainly used in the theory of non-associative algebras and appears in the construction of Lie algebras (cf. also Lie algebra; Non-associative rings and algebras).

A module $V$ over a field of characteristic not equal to two or three together with a trilinear mapping $( x , y , z ) \mapsto \langle x y z \rangle$ from $V \times V \times V$ to $V$ is said to be an Allison–Hein triple system (or a $J$-ternary algebra) if

\begin{equation} \tag{a1} \langle x y \langle u v w \rangle \rangle = \langle \langle x y u \rangle v w \rangle + \langle u \langle y x v \rangle w \rangle + \langle u v \langle x y w \rangle \rangle \end{equation}

\begin{equation} \tag{a2} \langle x y z \rangle - \langle z y x \rangle = \langle z x y \rangle - \langle x z y \rangle \end{equation}

for all $x , y , z , u , v , w \in V$.

From the identities (a1) and (a2) one deduces the relation

\begin{equation*} K ( \langle a b c ) , d ) + K ( c , \langle a b d \rangle ) + K ( a , K ( c , d ) b ) = 0, \end{equation*}

where $K ( a , b ) c = \langle a c b \rangle - \langle b c a \rangle$. Hence this triple system may be regarded as a variation of a Freudenthal–Kantor triple system. In particular, it is important that the linear span $\{ K ( a , b ) \} _ { \operatorname{span} }$ of the set $K ( a , b )$ is a Jordan subalgebra (cf. also Jordan algebra) of $( \text { End } V ) ^ { + }$ with respect to $A \circ B = ( A B + B A ) / 2$.

References

[a1] B.N. Allison, "A construction of Lie algebras from $J$-ternary algebras" Amer. J. Math. , 98 (1976) pp. 285–294
[a2] W. Hein, "A construction of Lie algebras by triple systems" Trans. Amer. Math. Soc. , 205 (1975) pp. 79–95
[a3] N. Kamiya, "A structure theory of Freudenthal–Kantor triple systems II" Commun. Math. Univ. Sancti Pauli , 38 (1989) pp. 41–60
[a4] K. Yamaguti, "On the metasymplectic geometry and triple systems" Surikaisekikenkyusho Kokyuroku, Res. Inst. Math. Sci. Kyoto Univ. , 306 (1977) pp. 55–92 (In Japanese)
How to Cite This Entry:
Allison-Hein triple system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Allison-Hein_triple_system&oldid=41899
This article was adapted from an original article by Noriaki Kamiya (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article