On a class of n-Leibniz deformations of the simple Filippov algebras

TL;DR

This paper investigates the deformation properties of simple finite-dimensional Filippov algebras, revealing rigidity for most cases and identifying a unique deformation for n=3.

Contribution

It proves that all simple finite-dimensional Filippov algebras with n>3 are rigid as n-Leibniz algebras, and identifies a non-trivial deformation for n=3.

Findings

All n>3 simple Filippov algebras are rigid as n-Leibniz algebras.

The n=3 simple Filippov algebras admit a non-trivial one-parameter deformation.

No non-trivial central extensions exist for these algebras as n-Leibniz algebras.

Abstract

We study the problem of the infinitesimal deformations of all real, simple, finite-dimensional Filippov (or n-Lie) algebras, considered as a class of n-Leibniz algebras characterized by having an n-bracket skewsymmetric in its n-1 first arguments. We prove that all n>3 simple finite-dimensional Filippov algebras are rigid as n-Leibniz algebras of this class. This rigidity also holds for the Leibniz deformations of the semisimple n=2 Filippov (i.e., Lie) algebras. The n=3 simple FAs, however, admit a non-trivial one-parameter infinitesimal 3-Leibniz algebra deformation. We also show that the $n\geq 3$ simple Filippov algebras do not admit non-trivial central extensions as n-Leibniz algebras of the above class.