Difference between revisions of "Lie triple system"
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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]]]. | 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 | + | 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> | <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> |
Revision as of 15:13, 19 March 2018
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.
References
[a1] | S. Helgason, "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press (1978) |
[a2] | N. Kamiya, S. Okubo, "On ![]() ![]() |
[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