Leibniz algebra deformations of a Lie algebra

TL;DR

This paper investigates Leibniz algebra deformations of a specific 3-dimensional nilpotent Lie algebra, revealing additional Leibniz deformations not present in the Lie algebra case and describing the versal deformation explicitly.

Contribution

It computes Leibniz algebra deformations of -dimensional nilpotent Lie algebra  and identifies three extra deformations beyond Lie deformations, providing the versal deformation base.

Findings

Identified 3 extra Leibniz deformations compared to Lie deformations.

Described the versal Leibniz deformation of -dimensional nilpotent Lie algebra.

Compared Leibniz and Lie deformations for the algebra .

Abstract

In this note we compute Leibniz algebra deformations of the 3-dimensional nilpotent Lie algebra $\mathfrak{n}_3$ and compare it with its Lie deformations. It turns out that there are 3 extra Leibniz deformations. We also describe the versal Leibniz deformation of $\mathfrak{n}_3$ with the versal base.