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$n$-Lie algebras. (English) Zbl 1210.17005
Summary: The notion of $n$-ary algebras, that is vector spaces with a multiplication concerning $n$-arguments, $n\ge 3$, 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 $\left(n\ge 3\right)$ 17B60 Lie (super)algebras associated with other structures