Cohomology, derivations and abelian extensions of 3-Lie algebras

Given a representation [Formula: see text] of a 3-Lie algebra [Formula: see text], we construct first-order cohomology classes by using derivations of [Formula: see text], [Formula: see text] and obtain a Lie algebra [Formula: see text] with a representation [Formula: see text] on [Formula: see text]. In the case that [Formula: see text] is given by an abelian extension [Formula: see text] of 3-Lie algebras with [Formula: see text], we obtain obstruction classes for extensibility of derivations of [Formula: see text] and [Formula: see text] to those of [Formula: see text]. An application of the representation [Formula: see text] to derivations is also discussed.