In mathematics, a Lie bialgebra is the Lie-theoretic case of a bialgebra: it's a set with a Lie algebra and a Lie coalgebra structure which are compatible.
It is a bialgebra where the comultiplication is skew-symmetric and satisfies a dual Jacobi identity, so that the dual vector space is a Lie algebra, whereas the comultiplication is a 1-cocycle, so that the multiplication and comultiplication are compatible. The cocycle condition implies that, in practice, one studies only classes of bialgebras that are cohomologous to a Lie bialgebra on a coboundary.
They are also called Poisson-Hopf algebras, and are the Lie algebra of a Poisson-Lie group.
Lie bialgebras occur naturally in the study of the Yang-Baxter equations.
Read more about Lie Bialgebra: Definition, Relation To Poisson-Lie Groups
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