n-Lie algebras

TL;DR

This paper explores the theory of n-Lie algebras, including their definitions, properties, and initial classification efforts, with a focus on nilpotent and filiform cases, and discusses their generalization to Strong Homotopy n-Lie algebras.

Contribution

It introduces the foundational notions of n-Lie algebras, studies specific classes like nilpotent and filiform n-Lie algebras, and extends the concept to Strong Homotopy n-Lie algebras.

Findings

Initial classification of nilpotent and filiform n-Lie algebras

Development of the theory of n-Lie algebras and their properties

Extension to Strong Homotopy n-Lie algebras

Abstract

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 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.