Cohomology and Deformation theory of crossed homomorphisms on Leibniz algebras

TL;DR

This paper develops a cohomology theory for crossed homomorphisms on Leibniz algebras using a differential graded Lie algebra framework, and explores their deformations and extendibility.

Contribution

It introduces a cohomology theory for crossed homomorphisms on Leibniz algebras via a new differential graded Lie algebra construction.

Findings

Constructed a differential graded Lie algebra for crossed homomorphisms.

Defined cohomology for crossed homomorphisms on Leibniz algebras.

Analyzed linear and formal deformations and their extendibility.

Abstract

In this paper, we construct a differential graded Lie algebra whose Maurer-Cartan elements are given by crossed homomorphisms on Leibniz algebras. This allows us to define cohomology for a crossed homomorphism. Finally, we study linear deformations, formal deformations and extendibility of finite order deformations of a crossed homomorphism in terms of the cohomology theory.