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 ![]() |
[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://encyclopediaofmath.org/index.php?title=Allison-Hein_triple_system&oldid=42985