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I once thought that the analogue of bialgebras and Lie bialegras is similar to that of (associative) algebras and Lie algebras, but it seems not that trivial.

Recall the definitions: a) bialgebra $A$ is a algebra $A$ with a comultiplication $\delta: A \to A\otimes A$ such that $\delta$ is coassociative and a morphism of algebras. b)Lie bialgebra $\mathfrak{g}$ is a Lie algebra $\mathfrak{g}$ with a cobracket $\Delta:\mathfrak{g} \to \mathfrak{g}\wedge\mathfrak{g}$ mathfrak{g}\otimes\mathfrak{g}$such that$\Delta$subjects to co-Jacobi identity and$\Delta$is a cocycle, where the action of$\mathfrak{g}$on$\mathfrak{g}\wedge\mathfrak{g}$\mathfrak{g}\otimes \mathfrak{g}$ is by adjoints.

Naïvely, we may expect the cobracket $\Delta$ to be a Lie algebra morphism but not a cocycle. Why so? This is the first part of my question, the other part: Is it possible to build a Lie bialgebra out of a bialgebra via alternating? Thanks in advance.

4 added 6 characters in body

I once thought that the analogue of bialgebras and Lie bialegras is similar to that of (associative) algebras and Lie algebras, but it seems not that trivial.

Recall the definitions: a) bialgebra $A$ is a algebra $A$ with a comultiplication $\delta: A \to A\otimes A$ such that $\delta$ is coassociative and a algebra morphism of algebras. b)Lie bialgebra $\mathfrak{g}$ is a Lie algebra $\mathfrak{g}$ with a cobracket $\Delta:\mathfrak{g} \to \mathfrak{g}\wedge\mathfrak{g}$ such that $\Delta$ subjects to co-Jacobi identity and $\Delta$ is a cocycle, where the action of $\mathfrak{g}$ on $\mathfrak{g}\wedge\mathfrak{g}$ is by adjoints.

Naïvely, we may expect the cobracket $\Delta$ to be a Lie algebra morphism but not a cocycle. Why so? This is the first part of my question, the other part: Is it possible to build a Lie bialgebra out of a bialgebra via alternating? Thanks in advance.

3 added 8 characters in body; added 18 characters in body; added 1 characters in body

I once thought that the analogue of bialgebras and Lie bialegras is similar to that of (associative) algebras and Lie algebras, but it seems not that trivial.

Recall the definitions: a) bialgebra $A$ is a algebra $A$ with a comultiplication $\delta: A \to A\otimes A$ such that $\delta$ is coassociative and a algebra morphism. b)Lie bialgebra $\mathfrak{g}$ is a Lie algebra $\mathfrak{g}$ with a cobracket $\Delta:\mathfrak{g} \to \mathfrak{g}\wedge\mathfrak{g}$ such that $\Delta$ subject subjects to co-Jacobi identity and $\Delta$ is a cocycle, where the action of $\mathfrak{g}$ on $\mathfrak{g}\wedge\mathfrak{g}$ is by adjoints.

Naïvely, we may expect the cobracket $\Delta$ to be a Lie algebra morphism but not cocycle. Why so? This is the first part of my question, the other part: Is it possible to build a Lie bialgebra out of a bialgebra via alternating? Thanks in advance.

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