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
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 .
|[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)|
Allison-Hein triple system. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Allison-Hein_triple_system&oldid=42985