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

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 3-Lie algebra to the semidirect product 3-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.