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 art 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 Delta(2). Examples include Stasheff's strongly homotopy Lie algebras and variants of homotopy Batalin-Vilkovisky algebras. There is a generalization with Delta replaced by an arbitrary odd derivation. We discuss applications and links with other constructions. (c) 2005 Published by Elsevier B.V.