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I read the two Wikipedia articles and it sounds like there is a relationship between the two, but I can't quite grasp it. They don't seem to be the same thing, but I can't demonstrate it in part because Lie coalgebra is defined using the exterior product, and Lie bialgebra is defined using tensor product. I'm confused :(

Can you tell me what is the difference between the two and how are they related? Is there a good text I could read?

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up vote 3 down vote accepted

A Lie coalgebra is a vector space $V$ endowed with a map $\delta:V\to\Lambda^2V$ satisfying some conditions.

A Lie bialgebra is a vector space $V$ endowen with a map $b:\Lambda^2V\to V$ and with a map $\delta:V\to\Lambda^2V$ satisfying some conditions, namely, that $(B,b)$ is a Lie algebra, that $(V,\delta)$ is a Lie coalgebra, and a compatibility condition between $b$ and $\delta$.

WIthout entering into details about what the actual conditions are, it should be clear that a Lie coalgebras and Lie bialgebras are different things! Additionally, it should also be clear that every Lie bialgebra is in particular a Lie coalgebra, simply by forgetting its map $b$. (On the other hand, a Lie coalgebra can be turned into a Lie bialgebra, by endowing it with a map $b$, in many ways in general---for example, one can always take $b=0$)

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