Infinitesimal deformations of null-filiform Leibniz superalgebras

TL;DR

This paper investigates the infinitesimal deformations of null-filiform Leibniz superalgebras, revealing their structure, classifications, and how single-generated algebras relate to these deformations within the algebraic variety.

Contribution

It characterizes the infinitesimal deformations of null-filiform Leibniz superalgebras and shows that single-generated Leibniz algebras are linear integrable deformations of a specific algebra.

Findings

The closure of orbits of single-generated Leibniz algebras forms an irreducible component.

Any single-generated Leibniz algebra is a linear integrable deformation of NF^{n}.

Results extend to Leibniz superalgebras with similar deformation properties.

Abstract

In this paper we describe the infinitesimal deformations of null-filiform Leibniz superalgebras over a field of zero characteristic. It is known that up to isomorphism in each dimension there exist two such superalgebras $NF^{n,m}$. One of them is a Leibniz algebra (that is $m=0$) and the second one is a pure Leibniz superalgebra (that is $m\neq 0$) of maximum nilindex. We show that the closure of union of orbits of single-generated Leibniz algebras forms an irreducible component of the variety of Leibniz algebras. We prove that any single-generated Leibniz algebra is a linear integrable deformation of the algebra $NF^{n}$. Similar results for the case of Leibniz superalgebras are obtained.