Deformations and cohomologies of embedding tensors on 3-Lie algebras

In this paper, first we introduce the notion of an embedding tensor on a 3-Lie algebra, which naturally induces a 3-Leibniz algebra. Using the derived bracket, we construct a Lie 3-algebra, whose Maurer-Cartan elements are embedding tensors. Consequently, we obtain the $L_{\infty}$-algebra that governs deformations of embedding tensors. We define the cohomology theory for embedding tensors on 3-Lie algebras. As applications, we show that if two formal deformations of an embedding tensor on a 3-Lie algebra are equivalent, then their infinitesimals are in the same cohomology class in the second cohomology group. Moreover, an order n deformation of an embedding tensor is extendable if and only if the obstruction class, which is in the third cohomology group, is trivial.