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Computational Methods for Biderivations of 4-dimensional nilpotent complex leibniz algebras

Ahmed Zahari Abdou, Bouzid Mosbahi

TL;DR

This paper develops computational methods to determine derivations, antiderivations, and biderivations of all $4$-dimensional nilpotent complex Leibniz algebras by using their classification. It provides explicit linear-algebraic algorithms that convert multilinear identities into systems of equations on matrix entries, enabling implementation in symbolic computation software. The authors report the dimensions of Der, AntiDer, and BiDer spaces across the 21 algebra types, ranging up to 7, 10, and 12 respectively, and discuss inner biderivations. The work offers practical tools for analyzing symmetry and invariants in low-dimensional Leibniz algebras and paves the way for higher-dimensional and deformation studies.

Abstract

This paper focuses on the biderivations of 4-dimensional nilpotent complex Leibniz algebras. Using the existing classification of these algebras, we develop algorithms to compute derivations, antiderivations, and biderivations as pairs of matrices with respect to a fixed basis. By utilizing computer algebra software such as Mathematica and Maple, we provide detailed descriptions and examples to illustrate these computations.

Computational Methods for Biderivations of 4-dimensional nilpotent complex leibniz algebras

TL;DR

This paper develops computational methods to determine derivations, antiderivations, and biderivations of all -dimensional nilpotent complex Leibniz algebras by using their classification. It provides explicit linear-algebraic algorithms that convert multilinear identities into systems of equations on matrix entries, enabling implementation in symbolic computation software. The authors report the dimensions of Der, AntiDer, and BiDer spaces across the 21 algebra types, ranging up to 7, 10, and 12 respectively, and discuss inner biderivations. The work offers practical tools for analyzing symmetry and invariants in low-dimensional Leibniz algebras and paves the way for higher-dimensional and deformation studies.

Abstract

This paper focuses on the biderivations of 4-dimensional nilpotent complex Leibniz algebras. Using the existing classification of these algebras, we develop algorithms to compute derivations, antiderivations, and biderivations as pairs of matrices with respect to a fixed basis. By utilizing computer algebra software such as Mathematica and Maple, we provide detailed descriptions and examples to illustrate these computations.
Paper Structure (5 sections, 2 theorems, 56 equations)

This paper contains 5 sections, 2 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Prelimieries
  3. An Algorithm for Finding Derivations
  4. An Algorithm for Finding AntiDerivations
  5. An Algorithm for Finding Biderivations

Key Result

Theorem 2.7

The isomorphism class of four-dimensional complex nilpotent Leibniz algebras given by the following representatives.

Theorems & Definitions (11)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Theorem 2.7
  • proof
  • proof
  • proof
  • ...and 1 more