Strongly homotopy Lie bialgebras and Lie quasi-bialgebras

Details Strongly homotopy Lie bialgebras and Lie quasi-bialgebras Strongly homotopy Lie bialgebras and Lie quasi-bialgebras

2006-01-12

arxiv.org/abs/math/0601301v4

Mathematics

Quantum Algebra

corrected version in Lett. Math. Physics

Scientific paper

10.1007/s11005-007-0167-x

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.

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