n-Lie algebras

By Michel Goze, Nicolas Goze, Elisabeth Remm. 2009

383 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Astronomy and Astrophysics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/07ff8e0991baf66943eea6ab3dd2189de0951b0c

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Citations in the corpus

M2 to D2 revisited (2008, methodology): “The multiple M2-brane model of Bagger-Lambert [1–3] and Gustavsson [4,5] is defined on Lie 3-algebras [6], which serve as the gauge symmetry algebras for the M2-brane world-volume theory.”
M5 from M2 (2008, neutral): “1 Introduction Recently, Bagger and Lambert [1–3] and Gustavsson [4,5] proposed a supersymmetric Lagrangian for multiple M2-branes with a Lie 3-algebra [6] as the internal symmetry L = − 1 2 〈DX , DμX 〉 + i 2 〈Ψ̄, ΓDμΨ〉 + i 4 〈Ψ̄, ΓIJ [X I , X , Ψ]〉 − V (X) + LCS , (1) where Dμ is the covariant deri”
Lie 3-Algebra and Multiple M2-branes (2008, background): “In the mathematical literature, the Lie 3-algebra (also known as Filippov algebra) is not new [9], and its structure has been studied to some extent.”
n-ary algebras: a review with applications (2010, methodology): “This last paper proved the analogue of the Levi decomposition for finitedimensional n-Lie algebras and showed that for n > 2 all (n + 1)-dimensional simple n-Lie algebras are of one type up to isomorphisms, the one given by Filippov [16], thus classifying the simple FAs.”
M5-brane in three-form flux and multiple M2-branes (2008, methodology): “In the BLG model, a background configuration of the M2-brane system corresponds to the choice of a Lie 3-algebra [4], and the Lie 3-algebra used for the M5brane is the Nambu-Poisson algebra [5] on a 3-manifold N which appears as the internal space from the M2-branes’ point of view, but it constitute”
Bagger-Lambert Theory for General Lie Algebras (2008, background): “The BL theory encodes the interactions of a three dimensional N = 8 multiplet, consisting of eight scalar fields X and their fermionic superpartners Ψ, and a non-propagating gauge (1)Known in the mathematical literature as 3-Lie algebras [5].”