# Allison-Hein triple system

The concept of a triple system, i.e. a vector space over a field together with a -trilinear mapping , 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 over a field of characteristic not equal to two or three together with a trilinear mapping from to is said to be an Allison–Hein triple system (or a -ternary algebra) if

(a1) |

(a2) |

for all .

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

where . 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 of the set is a Jordan subalgebra (cf. also Jordan algebra) of with respect to .

#### References

[a1] | B.N. Allison, "A construction of Lie algebras from -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) |

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Allison-Hein triple system.

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