\(n\)-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.


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