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m (Added that the standard embedding is Z2-graded)
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A [[triple system]] is a [[Vector space|vector space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130040/l1300401.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130040/l1300402.png" /> together with a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130040/l1300403.png" />-[[trilinear mapping]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130040/l1300404.png" />.
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A vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130040/l1300405.png" /> 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
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Out of 27 formulas, 26 were replaced by TEX code.-->
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130040/l1300407.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
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A [[triple system]] is a [[Vector space|vector space]] $V$ over a field $K$ together with a $K$-[[trilinear mapping]] $V \times V \times V \rightarrow V$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130040/l1300408.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130040/l1300409.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>
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\begin{equation} \tag{a1} [ x y z ] = - [ y x z ], \end{equation}
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130040/l13004010.png" />.
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\begin{equation} \tag{a2} [ x y z ] + [ y z x ] + [ z x y ] = 0, \end{equation}
  
Setting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130040/l13004011.png" />, then (a3) means that the left endomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130040/l13004012.png" /> is a derivation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130040/l13004013.png" /> (cf. also [[Derivation in a ring|Derivation in a ring]]). Thus one denotes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130040/l13004014.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130040/l13004015.png" />.
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\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}
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130040/l13004016.png" /> be a Lie triple system and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130040/l13004017.png" /> be the vector space of the direct sum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130040/l13004018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130040/l13004019.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130040/l13004020.png" /> is a Z2-graded [[Lie algebra|Lie algebra]] with respect to the product
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for all $x , y , z , u , v , w \in U$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130040/l13004021.png" /></td> </tr></table>
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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 $\operatorname{Inn} \, \operatorname{Der}A$.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130040/l13004022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130040/l13004023.png" />.
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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
  
This algebra is called the standard embedding Lie algebra associated with the Lie triple system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130040/l13004024.png" />. This implies that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130040/l13004025.png" /> 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]]].
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\begin{equation*} [ D + x , E + y ] : = [ D , E ] + D y - E x + L ( x , y ), \end{equation*}
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where $L ( x , y ) , D , E \in \operatorname { Inn } \operatorname { Der } A$, $x , y \in A$.
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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]].
 
Note that this kind of triple system is completely different from the combinatorial one of, e.g., a [[Steiner triple system]].
  
 
====References====
 
====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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130040/l13004026.png" />-Lie supertriple systems associated with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130040/l13004027.png" />-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>
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<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>

Revision as of 17:01, 1 July 2020

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 $U$ 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 $x , y , z , u , v , w \in U$.

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). 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
How to Cite This Entry:
Lie triple system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lie_triple_system&oldid=43183
This article was adapted from an original article by Noriaki Kamiya (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article