# Anti-Lie triple system

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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.

How to Cite This Entry:
Anti-Lie triple system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Anti-Lie_triple_system&oldid=16792
This article was adapted from an original article by Noriaki Kamiya (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article