# Anti-Lie triple system

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 |

**How to Cite This Entry:**

Anti-Lie triple system.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Anti-Lie_triple_system&oldid=42984