Infinitesimal deformations of naturally graded filiform Leibniz algebras

TL;DR

This paper investigates the infinitesimal deformations of naturally graded filiform Leibniz algebras, showing how all such algebras can be derived from these deformations and identifying a new rigid algebra.

Contribution

It extends the understanding of deformations of filiform Leibniz algebras and introduces a new rigid algebra within this class.

Findings

All n-dimensional filiform Leibniz algebras can be obtained via infinitesimal deformations of specific base algebras.

Describes the linear integrable deformations with a fixed basis of HL^2.

Identifies a new rigid algebra among the deformations.

Abstract

We describe infinitesimal deformations of complex naturally graded filiform Leibniz algebras. It is known that any $n$-dimensional filiform Lie algebra can be obtained by a linear integrable deformation of the naturally graded algebra $F_n^3(0)$. We establish that in the same way any $n$-dimensional filiform Leibniz algebra can be obtained by an infinitesimal deformation of the filiform Leibniz algebras $F_{n}^1,$ $F_{n}^2$ and $F_{n}^3(\alpha)$. Moreover, we describe the linear integrable deformations of above-mentioned algebras with a fixed basis of $HL^2$ in the set of all $n$-dimensional Leibniz algebras. Among these deformations we found one new rigid algebra.