n-Lie algebras
Michel Goze, Nicolas Goze, Elisabeth Remm

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 -ary algebras, that is vector spaces with a multiplication concerning -arguments, , became fundamental since the works of Nambu. Here we first present general notions concerning -ary algebras and associative -ary algebras. Then we will be interested in the notion of -Lie algebras, initiated by Filippov, and which is attached to the Nambu algebras. We study the particular case of nilpotent or filiform -Lie algebras to obtain a beginning of classification. This notion of -Lie algebra admits a natural generalization in Strong Homotopy -Lie algebras in which the Maurer Cartan calculus is well adapted.
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Taxonomy
TopicsAdvanced Topics in Algebra · Homotopy and Cohomology in Algebraic Topology · Advanced Differential Geometry Research
