Deformations, cohomologies and abelian extensions of compatible $3$-Lie algebras

In this paper, first we give the notion of a compatible $3$-Lie algebra and construct a bidifferential graded Lie algebra whose Maurer-Cartan elements are compatible $3$-Lie algebras. We also obtain the bidifferential graded Lie algebra that governs deformations of a compatible $3$-Lie algebra. Then we introduce a cohomology theory of a compatible $3$-Lie algebra with coefficients in itself and show that there is a one-to-one correspondence between equivalent classes of infinitesimal deformations of a compatible $3$-Lie algebra and the second cohomology group. We further study 2-order 1-parameter deformations of a compatible $3$-Lie algebra and introduce the notion of a Nijenhuis operator on a compatible $3$-Lie algebra, which could give rise to a trivial deformation. At last, we introduce a cohomology theory of a compatible $3$-Lie algebra with coefficients in arbitrary representation and classify abelian extensions of a compatible $3$-Lie algebra using the second cohomology group.