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-infinity-algebra whose Maurer-Cartan elements are crossed homomorphisms. Consequently, we obtain the twisted L-infinity-algebra that controls deformations of a given crossed homomorphism on 3-Lie algebras.