Strongly homotopy Lie bialgebras and Lie quasi-bialgebras

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