Nonabelian embedding tensors on 3-Lie algebras and 3-Leibniz-Lie algebras

In this paper, first we introduce the notion of a nonabelian embedding tensor on the 3-Lie algebra. Then, we introduce the notion of a 3-Leibniz-Lie algebra, which is the underlying algebraic structure of a nonabelian embedding tensor on the 3-Lie algebra, and can also be viewed as a nonabelian generalization of a 3-Leibniz algebra. Next we develop the cohomology of nonabelian embedding tensors on 3-Lie algebras with coefficients in a suitable representation and use the first cohomology group to characterize infinitesimal deformations. Finally, we investigate nonabelian embedding tensors on 3-Lie algebras induced by Lie algebras.