Abelian extensions of five-dimensional solvable Leibniz algebras

TL;DR

This work generalizes the central extension approach to solvable Leibniz algebras and applies it to classify one‑dimensional abelian extensions of five‑dimensional solvable algebras with a nontrivial three‑dimensional nilradical, as well as to the case where the nilradical is null‑filiform, for which a unique extension exists. The authors develop a cohomological framework using representations $(l,r)$ and $2$‑cocycles $Z^2(rak g,l,r)$ to construct extensions $rak g(oldsymbol{ abla})$, and they classify such extensions up to automorphisms by analyzing coboundaries $B^2(rak g,l,r)$. They obtain complete lists of nonisomorphic one‑dimensional abelian extensions for the four five‑dimensional base algebras with a 3D nilradical (H,L1,L2,L3), detailing explicit cocycle data and resulting extended multiplications, including several δ‑parametrized families. The results advance the understanding of solvable Leibniz algebra classifications and provide concrete extension models with potential applications in representation theory and algebraic structure theory.

Abstract

In this work, we extend the central extension method for solvable Leibniz algebras. Using this method, a complete classification of one-dimensional abelian extensions of five-dimensional solvable Leibniz algebras with a non-trivial three-dimensional nilradical is obtained. Furthermore, we explore extensions of solvable Leibniz algebras whose nilradical is null-filiform, establishing that, in this case, there exists a unique solvable abelian extension.

Theorems & Definitions (42)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Proposition 2.6
• Lemma 2.7
• proof
• Proposition 2.8
• proof