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  In mathematics, a Lie bialgebra is the Lietheoretic 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 skewsymmetric and satisfies a dual Jacobi identity, so that the dual vector space is a Lie algebra, whereas the comultiplication is a 1, 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.
 In mathematics, a Lie bialgebra is the Lietheoretic case of a bialgebra: it is a set with a Lie algebra and a Lie coalgebra structure which are compatible. It is a bialgebra where the comultiplication is skewsymmetric and satisfies a dual Jacobi identity, so that the dual vector space is a Lie algebra, whereas the comultiplication is a 1, 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.
 In mathematics, a Lie bialgebra is the Lietheoretic case of a bialgebra: it is a set with a Lie algebra and a Lie coalgebra structure which are compatible. It is a bialgebra where the multiplication is skewsymmetric and satisfies a dual Jacobi identity, so that the dual vector space is a Lie algebra, whereas the comultiplication is a 1, 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.

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  In mathematics, a Lie bialgebra is the Lietheoretic 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 skewsymmetric and satisfies a dual Jacobi identity, so that the dual vector space is a Lie algebra, whereas the comultiplication is a 1, 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 PoissonHopf algebras, and are the Lie algebra of a PoissonLie group. Lie bialgebras occur naturally in the study of the YangBaxter equations.
 In mathematics, a Lie bialgebra is the Lietheoretic case of a bialgebra: it is a set with a Lie algebra and a Lie coalgebra structure which are compatible. It is a bialgebra where the comultiplication is skewsymmetric and satisfies a dual Jacobi identity, so that the dual vector space is a Lie algebra, whereas the comultiplication is a 1, 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 PoissonHopf algebras, and are the Lie algebra of a Poisson–Lie group. Lie bialgebras occur naturally in the study of the Yang–Baxter equations.
 In mathematics, a Lie bialgebra is the Lietheoretic case of a bialgebra: it is a set with a Lie algebra and a Lie coalgebra structure which are compatible. It is a bialgebra where the multiplication is skewsymmetric and satisfies a dual Jacobi identity, so that the dual vector space is a Lie algebra, whereas the comultiplication is a 1, 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 PoissonHopf algebras, and are the Lie algebra of a Poisson–Lie group. Lie bialgebras occur naturally in the study of the Yang–Baxter equations.

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