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added more details to the definition at Lie-Poisson structure
have added some trivialities to Lie-Poisson structure:
added Properties – Poisson-Lie algebroid cohomology with the remark that in this case the Poisson bivector represents a trivial element in Lie algebroid cohomology.
There is also the entry Poisson-Lie group.
Well, all right then, I have moved the relevant paragraph there from the Idea-section to the examples section and added a cross-link here.
More serious is that we also have linear Poisson structure as a parallel entry stub, which is however synonymous with Lie-Poisson structure. I have cross-linked them for the moment, but eventually they ought to be merged.
Maybe it is not entirely synonymous in literature. For examples linear Poisson structures certainly do not include generalizations like for Lie algebroids. Second if you take a symplectic leaf not only the whole $\mathfrak{g}^*$, or any isomorphic symplectic manifold, or union of several such leaves, one still says linear Poisson structure while it does not complete to “Lie” thing. Third one also looks at the global bundle over a Lie group, like in the paper of Gutt, what is not the same as just $\mathfrak{g}^*$. But, surely all these are minor issues.
To me the title “Lie-Poisson structure” is more confusing, as I never know if they mean the Lie algebra or Lie group version.
The way to read is with brackets to the right:
a Lie-$\langle$Poisson structure$\rangle$ is a Poisson structure coming from a Lie bracket;
A Poisson-$\langle$Lie group$\rangle$ is a Lie group with Poisson data.
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