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 $\Delta:\mathfrak{g} \to \mathfrak{g}\otimes\mathfrak{g}$ such that $\Delta$ subjects to co-Jacobi identity and $\Delta$ is a cocycle, where the action of $\mathfrak{g}$ on $\mathfrak{g}\otimes \mathfrak{g}$ 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.