Topics on n-ary algebras

TL;DR

This paper explores the properties, cohomology, and rigidity of n-ary algebras, especially Filippov algebras, and discusses their Poisson counterparts and deformations, extending classical results to higher n.

Contribution

It extends Whitehead's lemma to all n-ary Filippov algebras, analyzes their cohomology, rigidity, and deformations, and connects them to n-Leibniz structures.

Findings

Semisimple FAs do not admit central extensions.

Filippov algebras are rigid for all n≥2.

Deformations of simple FAs retain skewsymmetry in certain brackets.

Abstract

We describe the basic properties of two n-ary algebras, the Generalized Lie Algebras (GLAs) and, particularly, the Filippov (or n-Lie) algebras (FAs), and comment on their n-ary Poisson counterparts, the Generalized Poisson (GP) and Nambu-Poisson (N-P) structures. We describe the Filippov algebra cohomology relevant for the central extensions and infinitesimal deformations of FAs. It is seen that semisimple FAs do not admit central extensions and, moreover, that they are rigid. This extends the familiar Whitehead's lemma to all $n\geq 2$ FAs, n=2 being the standard Lie algebra case. When the n-bracket of the FAs is no longer required to be fully skewsymmetric one is led to the n-Leibniz (or Loday's) algebra structure. Using that FAs are a particular case of n-Leibniz algebras, those with an anticommutative n-bracket, we study the class of n-Leibniz deformations of simple FAs that retain the skewsymmetry for the first n-1 entires of the n-Leibniz bracket.