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In the Wikipeadia article


on Lie coalgebras an author said: "... Just as the exterior algebra of vector fields on a manifold form a Lie algebra [...], the de Rham complex of differential forms on a manifold form a Lie coalgebra ...". Unfortunately the article gives no references at all.

If this is true I'm interested to learn more about that coalgebra structure on the de Rham complex of differential form and it would be great if someone knows a reference.

I'm especially interested in the interaction of that coalgebraic structure with calculus on differential forms (i.e exterior derivation, Lie derivation, insertion of vector fields ...)

Edit: As it turns out in the wiki on the exterior algebra is is said, that the wedge product and the shuffle coproduct together define a Hopf algebra and in particular the shuffle coproduct gives a coalgebra structure on the exterior algebra of a vector space.

In the light of this a reference to this Hopf algebra would be a reference of interest. But moreover it would be good to know how this Hopf algebra structure interacts with the calculus.

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Unfortunately the author "Silly rabbit" isn't a contributor to Wikipedia anymore. –  Mark.Neuhaus Mar 28 '12 at 6:08
Somebody really ought to fix the article. I have done a few edits but it is a drop in the ocean and I'm hesitant to do more. First, I think what the article refers to as "cocycle condition" has nothing to do with cocycles; instead, the "cocycle condition" is the name of a different axiom of Lie bi(!)algebras (an axiom which connects the Lie coalgebra and the Lie algebra structures, thus makes no sense for a Lie coalgebra alone). Second, exterior products seem to be abused. –  darij grinberg Mar 28 '12 at 6:26
But at least there is the coalgebra structure on the tensor power of the one forms. Leaves the question if this goes through to all exterior forms –  Mark.Neuhaus Mar 28 '12 at 8:36
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