Difference between revisions of "Lie triple system"
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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" />. | 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 | + | 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 Z2-graded [[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> | <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> |
Revision as of 11:35, 30 April 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 Z2-graded 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=42982