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A [[triple system]] is a [[Vector space|vector space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130250/a1302501.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130250/a1302502.png" /> together with a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130250/a1302503.png" />-[[trilinear mapping]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130250/a1302504.png" />. A triple system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130250/a1302505.png" /> satisfying
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<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/a/a130/a130250/a1302506.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
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<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/a/a130/a130250/a1302507.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</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]] $V \times V \times V \rightarrow V$. A triple system $V$ satisfying
  
<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/a/a130/a130250/a1302508.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/a/a130/a130250/a1302509.png" />, is called an anti-Lie triple system.
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\begin{equation} \tag{a2} \{ x y z \} + \{ y z x \} + \{ z x y \} = 0, \end{equation}
  
If instead of (a1) one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130250/a13025010.png" />, a [[Lie triple system|Lie triple system]] is obtained.
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\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}
  
Assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130250/a13025011.png" /> is an anti-Lie triple system and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130250/a13025012.png" /> is the [[Lie algebra|Lie algebra]] of derivations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130250/a13025013.png" /> containing the inner derivation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130250/a13025014.png" /> defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130250/a13025015.png" />. Consider <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130250/a13025016.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130250/a13025017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130250/a13025018.png" />, and with product given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130250/a13025019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130250/a13025020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130250/a13025021.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130250/a13025022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130250/a13025023.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130250/a13025024.png" />). Then the definition of anti-Lie triple system implies that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130250/a13025025.png" /> is a Lie [[Superalgebra|superalgebra]] (cf. also [[Lie algebra|Lie algebra]]). Hence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130250/a13025026.png" /> is an ideal of the Lie superalgebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130250/a13025027.png" />. One denotes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130250/a13025028.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130250/a13025029.png" /> and calls it the standard embedding Lie superalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130250/a13025030.png" />. This concept is useful to obtain a construction of Lie superalgebras as well as a construction of Lie algebras from Lie triple systems.
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for all $x , y , z , u , v \in V$, is called an anti-Lie triple system.
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If instead of (a1) one has $\{ x y z \} = - \{ y x z \}$, a [[Lie triple system|Lie triple system]] is obtained.
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Assume that $V$ is an anti-Lie triple system and that $\mathcal{D}$ is the [[Lie algebra|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|superalgebra]] (cf. also [[Lie algebra|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====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.R. Faulkner,  J.C. Ferrar,  "Simple anti-Jordan pairs"  ''Commun. Algebra'' , '''8'''  (1980)  pp. 993–1013</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  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</TD></TR></table>
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<table><tr><td valign="top">[a1]</td> <td valign="top">  J.R. Faulkner,  J.C. Ferrar,  "Simple anti-Jordan pairs"  ''Commun. Algebra'' , '''8'''  (1980)  pp. 993–1013</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  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</td></tr></table>

Latest revision as of 16:58, 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 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
How to Cite This Entry:
Anti-Lie triple system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Anti-Lie_triple_system&oldid=50248
This article was adapted from an original article by Noriaki Kamiya (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article