Biderivations of low-dimensional Leibniz algebras

TL;DR

This paper classifies low-dimensional Leibniz algebras based on their biderivations, providing a complete characterization for dimensions up to three and an explicit example for four dimensions, along with an algorithm for derivation computations.

Contribution

It offers a complete classification of biderivations for low-dimensional Leibniz algebras and introduces an algorithm for computing derivations and anti-derivations.

Findings

Classified Leibniz algebras of dimension up to three by their biderivations.

Computed biderivations for the four-dimensional Dieudonné Leibniz algebra.

Provided an algorithm to find derivations and anti-derivations as matrix pairs.

Abstract

In this paper we give a complete classification of the Leibniz algebras of biderivations of right Leibniz algebras of dimension up to three over a field $\mathbb{F}$, with $\operatorname{char}(\mathbb{F})\neq 2$. We describe the main properties of such class of Leibniz algebras and we also compute the biderivations of the four-dimensional Dieudonn\'e Leibniz algebra $\mathfrak{d}_1$. Eventually we give an algorithm for finding derivations and anti-derivations of a Leibniz algebra as pair of matrices with respect to a fixed basis.