View source for Lie triple system

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" />.

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

<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>

<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>

<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>

for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130040/l13004010.png" />.

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" />.

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 [[Lie algebra|Lie algebra]] with respect to the product

<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>

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" />.

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]]].

Note that this kind of triple system is completely different from the combinatorial one of, e.g., a Steiner triple system (cf. also [[Steiner system|Steiner 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 <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>