Cohomology and the controlling algebra of crossed homomorphisms on 3-Lie algebras

TL;DR

This paper develops a cohomology theory and deformation classification for crossed homomorphisms on 3-Lie algebras, linking them to Rota-Baxter operators and $L_ olinebreak_$-algebras, advancing the algebraic understanding of these structures.

Contribution

It introduces a new cohomology framework for crossed homomorphisms on 3-Lie algebras and constructs an $L_$-algebra controlling their deformations.

Findings

Established a characterization of crossed homomorphisms via homomorphisms to semidirect products.

Linked crossed homomorphisms to relative Rota-Baxter operators of weight 1.

Constructed an $L_$-algebra controlling deformations.

Abstract

In this paper, first we give the notion of a crossed homomorphism on a 3-Lie algebra with respect to an action on another 3-Lie algebra, and characterize it using a homomorphism from a Lie algebra to the semidirect product Lie algebra. We also establish the relationship between crossed homomorphisms and relative Rota-Baxter operators of weight 1 on 3-Lie algebras. Next we construct a cohomology theory for a crossed homomorphism on 3-Lie algebras and classify infinitesimal deformations of crossed homomorphisms using the second cohomology group. Finally, using the higher derived brackets, we construct an $L_\infty$-algebra whose Maurer-Cartan elements are crossed homomorphisms. Consequently, we obtain the twisted $L_\infty$-algebra that controls deformations of a given crossed homomorphism on 3-Lie algebras.