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Strongly Homotopy Lie Bialgebras and Lie Quasi-bialgebras

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Abstract

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.

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  1. Université de Lyon, Université Lyon 1, CNRS, UMR 5208 Institut Camille Jordan, 69622, Villeurbanne Cedex, France

    Olga Kravchenko

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  1. Olga Kravchenko
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Correspondence to Olga Kravchenko.

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This paper is dedicated to Jean-Louis Loday on the occasion of his 60th birthday with admiration and gratitude.

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Kravchenko, O. Strongly Homotopy Lie Bialgebras and Lie Quasi-bialgebras. Lett Math Phys 81, 19–40 (2007). https://doi.org/10.1007/s11005-007-0167-x

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