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 vector space $V$ with triple product $[\,,\,,\,]$ is said to be a Lie triple system if

\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 [ u v w ] ] = [ [ x y u ] v w ] + [ u [ x y v ] w ] + [ u v [ x y w ] ], \end{equation}

for all $x , y , z , u , v , w \in V$.

Setting $L ( x , y ) z : = [ x y z ]$, then (a3) means that the left endomorphism $L ( x , y )$ is a derivation of $V$ (cf. also Derivation in a ring). Thus one denotes $\{ L ( x , y ) \} _ { \text{span} }$ by $\operatorname{Inn} \, \operatorname{Der}A$.

Let $A$ be a Lie triple system and let $L ( A )$ be the vector space of the direct sum of $\operatorname{Inn} \, \operatorname{Der}A$ and $A$. Then $L ( A )$ is a Z2-graded Lie algebra with respect to the product

\begin{equation*} [ D + x , E + y ] : = [ D , E ] + D y - E x + L ( x , y ), \end{equation*}

where $L ( x , y ) , D , E \in \operatorname { Inn } \operatorname { Der } A$, $x , y \in A$.

This algebra is called the standard embedding Lie algebra associated with the Lie triple system $A$. This implies that $L ( A ) / \operatorname { Inn } \operatorname { Der } A$ is a homogeneous symmetric space (cf. also Homogeneous space; Symmetric space), that is, it is important in the correspondence with geometric phenomena and algebraic systems. The relationship between Riemannian globally symmetric spaces and Lie triple systems is given in [a4], and the relationship between totally geodesic submanifolds and Lie triple systems is given in [a1]. A general consideration of supertriple systems is given in [a2] and [a5].

Note that this kind of triple system is completely different from the combinatorial one of, e.g., a Steiner triple system.

References

[a1]	S. Helgason, "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press (1978)
[a2]	N. Kamiya, S. Okubo, "On $\delta$-Lie supertriple systems associated with $( \varepsilon , \delta )$-Freudenthal–Kantor supertriple systems" Proc. Edinburgh Math. Soc. , 43 (2000) pp. 243–260
[a3]	W.G. Lister, "A structure theory of Lie triple systems" Trans. Amer. Math. Soc. , 72 (1952) pp. 217–242
[a4]	O. Loos, "Symmetric spaces" , Benjamin (1969)
[a5]	S. Okubo, N. Kamiya, "Jordan–Lie super algebra and Jordan–Lie triple system" J. Algebra , 198 : 2 (1997) pp. 388–411