Allison-Hein triple system

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)