L∞L∞-algebra actions

We define the notion of action of an L∞L∞-algebra gg on a graded manifold MM, and show that such an action corresponds to a homological vector field on g[1]×Mg[1]×M of a specific form. This generalizes the correspondence between Lie algebra actions on manifolds and transformation Lie algebroids. In particular, we consider actions of gg on a second L∞L∞-algebra, leading to a notion of “semidirect product” of L∞L∞-algebras more general than those we found in the literature.