Embedding tensors on Lie $\infty$-algebras with respect to Lie $\infty$-actions

Abstract: Given two Lie $\infty$-algebras $E$ and $V$, any Lie $\infty$-action of $E$ on $V$ defines a Lie $\infty$-algebra structure on $E\oplus V$. Some compatibility between the action and the Lie $\infty$-structure on $V$ is needed to obtain a particular Loday $\infty$-algebra, the non-abelian hemisemidirect product. These are the coherent actions. For coherent actions it is possible to define non-abelian homotopy embedding tensors as Maurer-Cartan elements of a convenient Lie $\infty$-algebra. Generalizing the classical case, we see that a non-abelian homotopy embedding tensor defines a Loday $\infty$-structure on $V$ and is a morphism between this new Loday $\infty$-algebra and $E$.