Higher derived brackets and homotopy algebras

We give a construction of homotopy algebras based on higher derived brackets''. More precisely, the data include a Lie superalgebra with a projector on an Abelian subalgebra satisfying a certain axiom, and an odd element $\Delta$. Given this, we introduce an infinite sequence of higher brackets on the image of the projector, and explicitly calculate their Jacobiators in terms of $\Delta^2$. This allows to control higher Jacobi identities in terms of the order’’ of ${\mathrm{\Delta }}^2$. Examples include Stasheff’s strongly homotopy Lie algebras and variants of homotopy Batalin–Vilkovisky algebras. There is a generalization with $\text{D}$ replaced by an arbitrary odd derivation. We discuss applications and links with other constructions.