## $$n$$-Lie algebras.(English)Zbl 1210.17005

Summary: The notion of $$n$$-ary algebras, that is vector spaces with a multiplication concerning $$n$$-arguments, $$n\geq 3$$, became fundamental since the works of Nambu. Here we first present general notions concerning $$n$$-ary algebras and associative $$n$$-ary algebras. Then we are interested in the notion of $$n$$-Lie algebras, initiated by Filippov, and which is attached to the Nambu algebras. We study the particular case of nilpotent or filiform $$n$$-Lie algebras to obtain a beginning of classification. This notion of $$n$$-Lie algebra admits a natural generalization in strong homotopy $$n$$-Lie algebras in which the Maurer-Cartan calculus is well adapted.

### MSC:

 17A42 Other $$n$$-ary compositions $$(n \ge 3)$$ 17B60 Lie (super)algebras associated with other structures (associative, Jordan, etc.)
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