Difference between revisions of "Lie triple system"
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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" />. | + | 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 | 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 | ||
Revision as of 20:32, 19 September 2017
A triple system is a vector space 
over a field 
together with a 
-trilinear mapping 
.
A vector space 
with triple product 
is said to be a Lie triple system if
 | (a1) |
 | (a2) |
 | (a3) |
for all 
.
Setting 
, then (a3) means that the left endomorphism 
is a derivation of 
(cf. also Derivation in a ring). Thus one denotes 
by 
.
Let 
be a Lie triple system and let 
be the vector space of the direct sum of 
and 
. Then 
is a Lie algebra with respect to the product
 |
where 
, 
.
This algebra is called the standard embedding Lie algebra associated with the Lie triple system 
. This implies that 
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 (cf. also Steiner system).
References
| [a1] | S. Helgason, "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press (1978) |
| [a2] | N. Kamiya, S. Okubo, "On  -Lie supertriple systems associated with  -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 |
Lie triple system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lie_triple_system&oldid=41897