Strongly homotopy Lie bialgebras and Lie quasi-bialgebras

الأصل · EN

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∞-(quasi)bialgebra (strongly homotopy Lie (quasi)bialgebra). We recover an L∞-algebra structure as a particular case of our construction. The formal geometry interpretation leads to a definition of an L∞ (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∞-version of a Manin (quasi) triple and get a correspondence theorem with L∞-(quasi) bialgebras.