Cohomology and deformations of compatible 3-Lie superalgebras

Abstract

In this paper, we first introduce the notions of a compatible 3-Lie superalgebras and its representation. We construct a bidifferential graded Lie algebra whose Maurer–Cartan elements are compatible 3-Lie superalgebras. We also obtain the bidifferential graded Lie algebra which controls deformations of a compatible 3-Lie superalgebra. Then we investigate the cohomology theory of compatible 3-Lie superalgebras and consider the connection between the cohomology group of compatible 3-Lie superalgebras and the cohomology group of 3-Lie superalgebras. Furthermore, we develop the 1-parameter formal deformation theory of compatible 3-Lie superalgebras and prove that it is governed by the cohomology groups. At last, we study abelian extensions of compatible 3-Lie superalgebras and classify them by the second cohomology group.