View source for Allison-Hein triple system

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]]).

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

<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>

<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>

<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>

for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130200/a13020014.png" />.

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>

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" />.

====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>