On non-abelian extensions of Leibniz algebras

TL;DR

This paper classifies non-abelian extensions of Leibniz algebras using cohomology, constructs Leibniz 2-algebras from derivations, and describes extensions via Maurer-Cartan elements, advancing the algebraic understanding of Leibniz structures.

Contribution

It introduces a cohomological classification of non-abelian Leibniz extensions and links them to Leibniz 2-algebras and Maurer-Cartan elements.

Findings

Classified non-abelian extensions via second non-abelian cohomology.

Constructed Leibniz 2-algebras from derivations.

Connected extensions to Maurer-Cartan elements in a dg Lie algebra.

Abstract

In this paper, first we classify non-abelian extensions of Leibniz algebras by the second non-abelian cohomology. Then, we construct Leibniz 2-algebras using derivations of Leibniz algebras, and show that under a condition on the center, a non-abelian extension of Leibniz algebras can be described by a Leibniz 2-algebra morphism. At last, we give a description of non-abelian extensions in terms of Maurer-Cartan elements in a differential graded Lie algebra.