View source for Lie triple system

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

A vector space $U$ with triple product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130040/l1300406.png"/> 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 $x , y , z , u , v , w \in U$.

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|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|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|Homogeneous space]]; [[Symmetric 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 [[#References|[a4]]], and the relationship between totally geodesic submanifolds and Lie triple systems is given in [[#References|[a1]]]. A general consideration of supertriple systems is given in [[#References|[a2]]] and [[#References|[a5]]].

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

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">  S. Helgason,   "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press  (1978)</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  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</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  W.G. Lister,   "A structure theory of Lie triple systems"  ''Trans. Amer. Math. Soc.'' , '''72'''  (1952)  pp. 217–242</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  O. Loos,   "Symmetric spaces" , Benjamin  (1969)</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  S. Okubo,   N. Kamiya,   "Jordan–Lie super algebra and Jordan–Lie triple system"  ''J. Algebra'' , '''198''' :  2  (1997)  pp. 388–411</td></tr></table>