View source for Anti-Lie triple system

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

<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>

<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>

<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>

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.

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.

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.

====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>