Generalized representations, deformations and extensions of 3-Lie superalgebras

In this paper, we introduced generalized representations of 3-Lie superalgebras, which are associated to generalized semidirect product 3-Lie superalgebras. We also developed the corresponding cohomology theory and generalized one-parameter formal deformations. Furthermore, we proved that the infinitesimals and the extensibility of finite-order deformations of generalized one-parameter formal deformations are controlled by the new first and second cohomology groups, respectively. At last, we described split and non-split Abelian extensions by generalized semidirect products and Maurer-Cartan elements, respectively.