Difference between revisions of "Anti-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/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 | + | 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/a1302506.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table> |
Revision as of 20:32, 19 September 2017
A triple system is a vector space over a field
together with a
-trilinear mapping
. A triple system
satisfying
![]() | (a1) |
![]() | (a2) |
![]() | (a3) |
for all , is called an anti-Lie triple system.
If instead of (a1) one has , a Lie triple system is obtained.
Assume that is an anti-Lie triple system and that
is the Lie algebra of derivations of
containing the inner derivation
defined by
. Consider
with
and
, and with product given by
,
,
for
,
(
). Then the definition of anti-Lie triple system implies that
is a Lie superalgebra (cf. also Lie algebra). Hence
is an ideal of the Lie superalgebra
. One denotes
by
and calls it the standard embedding Lie superalgebra of
. This concept is useful to obtain a construction of Lie superalgebras as well as a construction of Lie algebras from Lie triple systems.
References
[a1] | J.R. Faulkner, J.C. Ferrar, "Simple anti-Jordan pairs" Commun. Algebra , 8 (1980) pp. 993–1013 |
[a2] | 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 |
Anti-Lie triple system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Anti-Lie_triple_system&oldid=41898