Anti-Lie triple system

A triple system is a vector space $V$ over a field $K$ together with a $K$-trilinear mapping $V \times V \times V \rightarrow V$. A triple system $V$ satisfying

\begin{equation} \tag{a1} \{ x y z \} = \{ y x z \}, \end{equation}

\begin{equation} \tag{a2} \{ x y z \} + \{ y z x \} + \{ z x y \} = 0, \end{equation}

\begin{equation} \tag{a3} \{ x y \{ z u v \} \} = \{ x y z \} u v \} + \{ z \{ x y u \} v \} + \{ z u \{ x y v \} \}, \end{equation}

for all $x , y , z , u , v \in V$, is called an anti-Lie triple system.

If instead of (a1) one has $\{ x y z \} = - \{ y x z \}$, a Lie triple system is obtained.

Assume that $V$ is an anti-Lie triple system and that $\mathcal{D}$ is the Lie algebra of derivations of $V$ containing the inner derivation $L$ defined by $L ( x , y ) z = \{ x y z \}$. Consider ${\cal L} = L _ { 0 } \oplus L_1$ with $L _ { 0 } = \mathcal{D}$ and $L _ { 1 } = V$, and with product given by $[ a _ { 1 } , a _ { 2 } ] = L ( a _ { 1 } , a _ { 2 } ) \in L ( V , V )$, $- [ a _ { 1 } , D _ { 1 } ] = [ D _ { 1 } , a _ { 1 } ] = D _ { 1 } a _ { 1 }$, $[D _ { 1 } , D _ { 2 } ] = D _ { 1 } D _ { 2 } - D _ { 2 } D _ { 1 } \in \mathcal{D}$ for $a _ { i } \in V$, $D _ { i } \in \mathcal{D}$ ($i = 1,2$). Then the definition of anti-Lie triple system implies that $\mathcal{L}$ is a Lie superalgebra (cf. also Lie algebra). Hence $L ( V , V ) \oplus V$ is an ideal of the Lie superalgebra $\mathcal{L} = \mathcal{D} \oplus V$. One denotes $L ( V , V ) \oplus V$ by $L ( V )$ and calls it the standard embedding Lie superalgebra of $V$. This concept is useful to obtain a construction of Lie superalgebras as well as a construction of Lie algebras from Lie triple systems.

References

[a1]	J.R. Faulkner, J.C. Ferrar, "Simple anti-Jordan pairs" Commun. Algebra , 8 (1980) pp. 993–1013
[a2]	N. Kamiya, "A construction of anti-Lie triple systems from a class of triple systems" Memoirs Fac. Sci. Shimane Univ. , 22 (1988) pp. 51–62