[Paper Review] The classification of 4-dimensional Leibniz algebras

E.M. Cañete, Abror Khudoyberdiyev|arXiv (Cornell University)|Jan 31, 2013

Advanced Topics in Algebra8 references4 citations

TL;DR

This paper classifies all complex 4-dimensional solvable Leibniz algebras by determining their structure through the classification of non-nilpotent outer derivations of their nilradicals. It identifies 44 non-isomorphic algebras, including 14 new ones, extending prior work on nilpotent 4D Leibniz algebras and providing a complete classification of the solvable case.

ABSTRACT

This paper is a contribution to the development of the non associative algebras theory. More precisely, this work deals with the classification of the complex 4-dimensional Leibniz algebras. Note that the classification of 4-dimensional nilpotent complex Leibniz algebras was obtained in [1]. Therefore we will only consider non nilpotent case in this work.

Motivation & Objective

• To complete the classification of 4-dimensional complex Leibniz algebras by focusing on the non-nilpotent solvable case, following prior classification of nilpotent 4D Leibniz algebras.
• To extend the method used for solvable Lie algebras—applying non-nilpotent outer derivations of the nilradical—to the non-Lie setting of Leibniz algebras.
• To systematically determine all isomorphism classes of 4D solvable Leibniz algebras with specified nilradicals, particularly λ₅ and λ₆.
• To verify the non-isomorphism of the constructed algebras using a custom Mathematica-based algorithm.
• To provide a comprehensive list of 44 non-isomorphic 4D solvable Leibniz algebras, including 14 new algebras not previously known.

Proposed method

• Utilized the Levi-Malcev-type decomposition for Leibniz algebras, focusing on solvable algebras with given nilradicals.
• Applied the method of classifying solvable Leibniz algebras via non-nilpotent outer derivations of the nilradical, adapted from Lie algebra techniques.
• Constructed algebras by defining bracket structures using derivations represented as matrices, particularly analyzing the right multiplication operator R_x.
• Imposed constraints from the Leibniz identity and nilpotency conditions on the derivations to ensure algebra consistency.
• Used basis transformations and normalization to eliminate redundant parameters and reduce isomorphic copies.
• Implemented a computational algorithm in Mathematica to verify that the constructed algebras are pairwise non-isomorphic.

Experimental results

Research questions

• RQ1 What are all the isomorphism classes of 4-dimensional complex solvable Leibniz algebras?
• RQ2 How can the classification of solvable Leibniz algebras be systematically achieved using outer derivations of the nilradical?
• RQ3 Which 4D solvable Leibniz algebras arise from nilradicals isomorphic to λ₅ and λ₆?
• RQ4 How can one distinguish between non-isomorphic Leibniz algebras in the 4D case?
• RQ5 What new algebras emerge in the classification that were not previously known?

Key findings

• The paper presents a complete classification of 4-dimensional complex solvable Leibniz algebras, identifying 44 non-isomorphic algebras.
• Among the 44 algebras, 14 are new and not previously classified, including algebras such as 𝒮₄₃ and 𝒮₄₄ with specific bracket structures.
• The classification is achieved by analyzing non-nilpotent outer derivations of the nilradicals λ₅ and λ₆, which are the only possible nilradicals for 4D solvable Leibniz algebras.
• The algebra 𝒮₄₃ is constructed with nilradical λ₆ and bracket relations involving [e₁,e₁]=e₂, [e₁,x]=e₁, [e₂,x]=2e₁, [e₃,x]=3e₁.
• The algebra 𝒮₄₄ is constructed with nilradical λ₅ and bracket relations including [e₁,e₂]=e₃, [e₁,x]=e₁, [e₂,x]=e₂, [e₃,x]=2e₃.
• A computer-assisted verification using a Mathematica-based algorithm confirms that all 44 algebras are pairwise non-isomorphic.