Structures of Lie algebras, Lie coalgebras, Lie bialgebras and Lie quasibialgebras are presented as solutions of MaurerâCartan equations on corresponding governing differential graded Lie algebras using the big bracket construction of KosmannâSchwarzbach. This approach provides a definition of an L â-(quasi)bialgebra (strongly homotopy Lie (quasi)bialgebra). We recover an L â-algebra structure as a particular case of our construction. The formal geometry interpretation leads to a definition of an L â (quasi)bialgebra structure on V as a differential operator Q on V, self-commuting with respect to the big bracket. Finally, we establish an L â-version of a Manin (quasi) triple and get a correspondence theorem with L â-(quasi)bialgebras.