The notion of n-ary algebras, that is vector spaces with a multiplication concerning n-arguments, n ≥ 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 will be 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.This work has been presented during the 11 eme Rencontre Nationale de Géométrie Différentielle et Applications RNGDA11, Faculté des Sciences Ben-Msik, Casablanca. This meeting was dedicated to Professor Younes BENSAID, Member of the Académie Française de Chirurgie.1 n-ary algebras Basic definitionsLet K be a commutative field of characteristic zero and V a K-vector space. Let n be in N, n ≥ 2. A n-ary algebra structure on V is given by a linear mapWe denote by (V, µ) such an algebra. Classical algebras (associative algebras, Lie algebras, Leibniz algebras for example) are binary that is given by a 2-ary product. In this paper, we are interested in the study of n-ary algebras for n ≥ 3. A subalgebra of the n-ary algebra (V, µ) is a vector subspace W of V such that the restriction of µ to W ⊗n satisfies µ(W ⊗n ) ⊂ W. In this case (W, µ) is also a n-ary algebra.

A Lie-admissible algebra gives a Lie algebra by anticommutativity. In this work we describe remarkable types of Lie-admissible algebras such as Vinberg algebras, pre-Lie algebras or Lie algebras. We compute the corresponding binary quadratic operads and study their duality. Considering Lie algebras as Lie-admissible algebras, we can define for each Lie algebra a cohomology with values in an Lie-admissible module. This leads to the study some deformations of Lie algebras in the classes of Lie-admissible algebras.

Abstract. We examine the problem of describing filiform Lie algebras over the field C admitting a semisimple derivation. As applications of obtained results we give a description of Lie algebras with a given filiform nilradical.

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