Cohomology, superderivations, and abelian extensions of $3$-Lie superalgebras

Abstract

The main object of study of this paper is the notion of 3-Lie superalgebras with superderivations. We consider a representation (Φ, P) of a 3-Lie superalgebra Q on P and construct first-order cohomologies by using superderivations of P, Q which induces a Lie superalgebra TΦ and its representation Ψ. Then we consider abelian extensions of 3-Lie superalgebras of the form 0 → P ֒→ L → Q → 0 with [P, P, L] = 0 and construct an obstruction class to extensibility of a compatible pair of superderivation. Moreover we prove that a pair of superderivation is extensible if and only if its obstruction class is trivial under some suitable conditions.